A summation on Bernoulli numbers

Size: px
Start display at page:

Download "A summation on Bernoulli numbers"

Transcription

1 Journal of Number Theory 111 ( A summaton on Bernoull numbers Kwang-Wu Chen Department of Mathematcs and Computer Scence Educaton, Tape Muncpal Teachers College, No. 1, A-Kuo West Road, Tape, Tawan 100, Republc of Chna Receved December 003; revsed 5 Aprl 004 Communcated by A. Granvlle Avalable onlne 7 December 004 Abstract In ths paper, our am s to nvestgate the summaton form of Bernoull numbers B n, such as n k=0 ( nk Bk+m. We derve some basc denttes among them. These numbers can form a Sedel matrx. The upper dagonal elements of ths Sedel matrx are called the medan Bernoull numbers. We determne the prme dvsors of ther numerators and denomnators. And we characterze ther ordnary generatng functon as the unque soluton of some functonal equaton. At last, we also obtan the contnued fracton representaton of ther ordnary generatng functon and ther value of Hankel determnant. 004 Elsever Inc. All rghts reserved. MSC: prmary 11B68; 15A15 Keywords: Medan Bernoull numbers; Sedel matrx; Hankel determnant 1. Introducton The Bernoull numbers B n are defned by the recurrence relaton m =1 ( m B m = 0 (m, B 0 = 1. E-mal address: kwchen@tmtc.edu.tw X/$ - see front matter 004 Elsever Inc. All rghts reserved. do: /j.jnt

2 K.-W. Chen / Journal of Number Theory 111 ( Thus we have B 0 = 1, B 1 = 1, B = 1 6, B 3 = 0, and so on. The Bernoull numbers have extensve applcatons on many areas. One of remarkable propertes s related to the values at ntegers of the Remann zeta functon ζ(s, such as ζ(n = ( 1 n+1 (πn B n, ζ(1 n = B n (n! n for postve ntegers n. Recently, Akyama and Tangawa [1] expressed the values ζ k (0,...,0, n and ζ k ( n, 0,...,0 of the Euler Zager s multple zeta functon, whch s defned by ζ k (s 1,...,s k = 1 n s 1 0<n 1 <n < <n k 1 n s n s k k as a lnear combnaton of B n+ for 1 k (See [1, Eqs. (6, (10]. However, the summaton formulae for these knds are very few n the lterature. We lst two man denttes of these knds (Refs. [6,9] n+1 ( n + 1 (k + n + 1B k+n = 0 (1 k k=0 and m k=0 ( m B k+n = ( 1 m+n k k=0 ( n B k+m. ( k In order to nvestgate the summatons of the form m k=0 ( m k Bk+n, the notaton of a Sedel matrx wll be llustrated n the followng: The Sedel matrx (a n,k n,k 0 assocated wth the ntal sequence a 0,n s defned by [ 5] a n,k = a n 1,k + a n 1,k+1 (3 or, equvalently, by a n,k = ( n a 0,k+. (4

3 374 K.-W. Chen / Journal of Number Theory 111 ( We usually denote the exponental generatng functon of the ntal sequence a 0,n as A(x = a 0,n x n. n! The ordnary generatng functon of the ntal sequence a 0,n s denoted as a(x = a 0,n x n+1. The ordnary generatng functon a(x s, n a formal sense, the Laplace transform of A(x, that s, 0 A(te t/x dt = a(x. The Bernoull polynomals B n (x are defned by te xt e t 1 = B n (xt n. n! It s easly to get B n = B n (0. The Sedel matrx (a n,k n,k 0 obtaned by takng the sequence of Bernoull numbers as the ntal sequence s represented as follows (Ref. Matrx (ES1 n [3]. 1 1/ 1/6 0 1/30 0 1/ / 1/3 1/6 1/30 1/30 1/4 1/4... 1/6 1/6 /15 1/15 1/105 1/ /30 1/15 8/105 4/ /30 1/30 1/105 4/ /4 1/1... 1/4 1/ Therefore we take a 0,n = B n n Eq. (4. Then a n,k = ( n B k+. We denote ths Sedel matrx as the BS-matrx. Some basc propertes of the BS-matrx gve Eqs. (1 and (. We wll provde more detal later n Secton.

4 K.-W. Chen / Journal of Number Theory 111 ( Three mportant sequences n the Sedel matrx are the ntal sequence (a 0,n, the fnal sequence (a n,0, and the upper dagonal sequence (a n,n+1. The medan Genocch numbers H n+1 (Refs. [3 5] could be defned by the upper dagonal elements of the Sedel matrx assocated wth the ntal sequence as the Genocch numbers G n, whch s defned by t e t + 1 = n=1 G n t n. n! Dumont [4] defned the medan Euler numbers R n, L n as the upper dagonal elements of the Sedel matrx assocated wth the ntal sequence as the Euler numbers E n, ( 1 n+1 E n, respectvely, whch s defned by sech (x + tanh(x = E n x n. n! Snce the ntal sequence B n and the fnal sequence B n (1 = ( 1 n B n of the BSmatrx are both well-known, n ths paper, we focus our attenton on the upper dagonal sequence, whch we call the medan Bernoull numbers, K n, n accord wth the above nomenclature n [4,5]. It can be seen that the man dagonal elements are just negatve twce of the upper dagonal elements. And ths wll be proved n Secton. Let p be a prme number. We call a ratonal number p-ntegral f ts denomnator s not dvded by p. Ifa and b are ratonal numbers, then by a b(mod p r we mean that (a b/p r s p-ntegral. For example, 1 5 (mod. We defne ord p(a to be the largest nteger for whch a/p ord p(a s p-ntegral; so ord ( 1 = 1 and ord ( 4 3 =. In Secton 3, we determne the prme dvsors of the denomnators D n of the medan Bernoull numbers K n and the order of of the numerators N n of K n. Tables 1 and gve the prme factorzatons of N n and D n for 0 n 30. Theorem 1.1. (1 The denomnators D n are square-free numbers. ( For a postve nteger n, the set of the all odd prme dvsors of D n s { p : odd prme n m (3 If s a dvsor of D n, then n = 0 or 1. (4 For n 1, ord (N n = [ n 1 p 1 n } m 1, m N+. ], where [x] means the largest nteger n x. In Secton 4, we characterze the ordnary generatng functon of K n as the unque soluton of some functonal equaton.

5 376 K.-W. Chen / Journal of Number Theory 111 ( Table 1 Medan Bernoull numerators N n factored n Prme factorzaton of N n Theorem 1.. Let m(x = K n x n+1 = 1 x x 1 15 x x x x6 (5 be the ordnary generatng functon of K n. Then m(x s the unque soluton of the functonal equaton x x ( x m + x + ( x m = x. (6 1 x x 1 + x

6 K.-W. Chen / Journal of Number Theory 111 ( Table Medan Bernoull denomnators D n factored n Prme factorzaton of D n In the last secton, we obtan the contnued fracton representaton of m(x, and we also obtan the Hankel determnant det 0,j n ( K+j. Theorem 1.3. For any non-negatve nteger n, the Hankel determnant of the medan Bernoull numbers s ( ( det K+j = 1 n+1 n 0,j n =1 ( ( n +1 (4 3(4 1. (7 (4 + 1

7 378 K.-W. Chen / Journal of Number Theory 111 ( Some basc relatons Consder the Sedel matrx (a n,k n,k 0 wth the ntal sequence B n. From Eq. (4 we have ( n a n,k = B k+. (8 Proposton.1. For k, n 0, a k,n = ( 1 k+n a n,k. (9 Proof. Ths result s followed on usng nducton on the frst ndex k n a k,n and the recursve relaton for a n,k. Substtute a n,k by Eq. (8 n Eq. (9, then we get Eq. (. Corollary.. For k, n 0, k ( k B n+ = ( 1 k+n From Proposton.1, t s clearly that a n,n+1 = a n+1,n. Hence ( n B k+. (10 a n,n = a n+1,n a n,n+1 = a n,n+1 = K n. (11 Therefore the man dagonal elements are just negatve twce of the upper dagonal elements. In the followng we gve the proof of Eq. (1. Corollary.3. For any non-negatve nteger n, we have n+1 ( n + 1 ( + n + 1B +n = 0. (1 Proof. Eq. (11 s equvalent to 1 n + 1 ( n B n+ = n+1 =1 ( n + 1 (n B n+ = B n+1. ( n B n+, 1 Multply (n + 1 on both sdes, then we get the desred dentty.

8 K.-W. Chen / Journal of Number Theory 111 ( Let k = n n Eq. (3 and we apply Eq. (11, then we have a recursve relaton wth K n. n 1 ( n 1 K n = K n 1 + B n++1. (13 Solvng ths recursve relaton we get, for n 1, n ( n 1 K n = B + j=0 j+1 ( j ( j + 1 B +j+3. (14 In the end of ths secton we lst two expressons of K n whch can be easly got from Eqs. (8 and (11. For n 0, K n = ( n B n+1+ (15 and K n = 1 ( n B n+. (16 3. Prme dvsors of the D n and N n We frst nvestgate the order of of K n. Proposton 3.1. For n and m n + 1, [ ] n 1 ord (a n,m =. (17 And ord (a 1,m = 1, for m. Proof. We use nducton on n. For n = 1 and m, a 1,m = B m + B m+1. Hence a 1,k = a 1,k 1 = B k. As a result ths mples ord (a 1,m = 1. For n = and m 3, we have a,k = B k + B k+ and a,k 1 = B k for k. Snce B t 1 (mod 4 for t (Ref. [8, p. 47], t could get B k + B k+ 1 (mod. Thus

9 380 K.-W. Chen / Journal of Number Theory 111 ( ord (a,k = 0. On the other hand, a,k 1 = B k 1 (mod 4, for k. Therefore ord (a,k 1 = 0. Assume that for n 3 and k<n, ord (a k,m = [ k 1 ] wth m k + 1. Then for m n + 1 a n,m = a n 1,m + a n 1,m+1 = a n,m + a n,m+ + a n,m+1. Now by the nducton hypothess we have [ ] n 1 ord (a n,m + a n,m+ [ ] n 1 and ord (a n,m+1 =. Ths completes our proof. Corollary 3.. For n we have [ ] n 1 ord (K n =. (18 The nformaton of Eqs. (9, (11, and Proposton 3.1 have been collected, the value of ord (a n,m for all n, m can be easly predcted. To descrbe the denomnators of K n we need the von Staudt Claussen Theorem: Theorem 3.3 (Ireland and Rosen [8, p. 33]. For m 1, B m = A m 1 p, (19 p 1 m where A m Z and the sum s over all prmes p such that p 1 m. Proposton 3.4. Let q be a prme number. Then for any non-negatve nteger n, qk n s q-ntegral and the denomnator D n of K n s a square-free number. Proof. Gven the fact that qb k s q-ntegral for any k and from the result of Eq. (15, we know that qk n s q-ntegral and therefore D n s square-free.

10 K.-W. Chen / Journal of Number Theory 111 ( Combne Eqs. (15 and (19, we can rewrte K n as K n = ( n B n+1+ m ( m A m+ 1 =1 = m 1 ( m 1 A m+ p 1 m+ p 1 m+ 1 f n = m, p 1 f n = m 1. p (0 It can be seen that the largest possble prme dvsor of D n s n + 1. Now we need a result whch s proved by Glasher n Lemma 3.5 (Granvlle [7]. For any gven prme q and ntegers 1 j,k q 1, we have 1 m n m j (mod q 1 ( n m for all postve ntegers n k(mod q 1. ( k (mod q (1 j Let q be a fxed odd prme number. Now we need only to know whether q s a dvsor of D n or not, for q 1 3q 5 n. Proposton 3.6. Gven a fxed odd prme q, f q 1 n q 1, then q s a dvsor of D n ; f q n 3q 5, then q s not a dvsor of D n. Proof. From Eq. (0 we need to consder whether the followng two factors: m =1 p 1 m+ ( m 1 p for n = m and m 1 p 1 m+ ( m 1 p for n = m 1

11 38 K.-W. Chen / Journal of Number Theory 111 ( are q-ntegral or not. Here, we frst deal wth the case for n = m. The method for the remanng case, n = m 1, s smlar, so we omt t. n q 1, then ths mples If q 1 So q cannot be a dvsor of ( m 1. Let n<n+ 1 q n + 1. C = { 1 m q 1 s a dvsor of m + }. Snce n q 1 n, there exsts an unque nteger 0 q m wth q 1 = m + q. If 1 q m, then q C. If q = 0, then q 1 = m s a dvsor of 4m. And = m C. Therefore the set C s not empty. If C contans at least two dstnct elements, then ths wll force the number q 1 s less than n. Ths contradcton ensures a fact that the number of the elements n C s one. The prme number q only occur once n the summaton m =1 p 1 m+ ( m 1 p, therefore q s a dvsor of D n. Now we assume that q n 3q 5. We can rewrte n = q + j, for 0 j q 5 If 1 q, then we have n + q + 1 m + n = q + j, q + 1 m + 3q 5. Dvdng (q 1 from m +, we have a remander r wth 3 r q 3. Thus q 1 s not a dvsor of m +. On the other hand, f m + 1 q, then we use the same trck and we also can prove that q 1 s not a dvsor of m +. Therefore f q 1 s a dvsor of m +, for some. Then both 1 and m + 1 must be less than q, and ths mples q ( m 1. Ths completes our proof. Now we come to the results and the followng theorem can be derved from them. Theorem 3.7. For a postve nteger n, the set of the all odd prme dvsors of D n s { p : odd prme n p 1 n },m N+. ( m m 1.

12 K.-W. Chen / Journal of Number Theory 111 ( Proof. For a fxed odd prme q, q s a dvsor of D m+k(q 1, for q 1 m q 1 and k 0. Hence for a fxed postve nteger n, wehave That s for k 0, q 1 + k(q 1 n = m + k(q 1 q 1 + k(q 1. n k + 1 q 1 n k Ordnary generatng functon and functonal equaton Proposton 4.1 (Dumont [4, Proposton 6]. Gven a Sedel matrx (a n,k n,k 0, then the ordnary generatng functon of ts ntal sequence, of ts man dagonal, and of ts upper dagonal, respectvely, denoted by a(x = a 0,n x n+1, d 0 (x = a n,n x n, d 1 (x = a n,n+1 x n+1 satsfy the dentty and Let m(x = b(x = a(x = x d 0 ( x 1 + x + d 1 ( x 1 + x. (3 B n x n+1 = x 1 x x x x7 K n x n+1 = 1 x x 1 15 x x x5. That s, b(x s the ordnary generatng functon of the ntal sequence B n, and m(x s the ordnary generatng functon of the upper dagonal sequence K n. Now we combne Eq. (11 and Proposton 4.1 and we get the relaton between these two functons n the followng theorem.

13 384 K.-W. Chen / Journal of Number Theory 111 ( Theorem 4.. We have ( b(x = 1 + ( x m. (4 x 1 + x Corollary 4.3. For k 1, we have k ( k n k n k n K n = B k, (5 k 1 ( k 1 n k 1 n k 1 n K n = B k 1. (6 Proof. Ths follows by extractng the coeffcent of x n n Eq. (4. The two equatons n the above corollary can be rewrtten as [ n ] ( n m n m n m K m = (δ 1n 1B n n 1. (7 m=0 Theorem 4.4. Let (x be a formal power seres. Then the followng three assertons are equvalent: t (1 (x = 0 snh(x e t/x dt; ( x ( (x and + x are both odd; ( 1 + ( x x x (3 = x. 1 x 1 + x Proof. ((1 (: Let f(x= 0 e t/x F(tdt. Recall that 0 e t/x t n dt = n!x n+1, so f(x s even (resp. odd, f and only f F(x s odd (resp. even. Note that Snce t snh t ( x + x = 1 + x 0 t coth te t/x dt. and t cosh t are both even, asserton ( follows mmedately.

14 K.-W. Chen / Journal of Number Theory 111 ( ( x (( (3: Snce + x s odd, we have 1 + x ( ( x x + x = x. 1 x 1 + x Asserton (3 follows from the fact that (x s odd. ((3 (1: Let (x = 0 e t/x F(tdt. Then ( x x = = 0 1 x ( x 1 + x e t/x F (t(e t e t dt. But x = 0 e t/x dt; hence F (t(e t e t = t and F(t = t snh t. t e t e t = Theorem 4.5. The followng denttes hold: ( x b(x =, (8 x + (x = 1 ( 4x x m 1 x. (9 Proof. Accordng to Theorem 4., t suffces to prove Eq. (8. Note that So te t snh t = t e t 1 = n B n t n. n! ( x = 1 x + 1 n+1 B n x n+1, whch clearly mples Eq. (8. The next corollary s an mmedate consequence of Theorem 4.4 and Theorem 4.5.

15 386 K.-W. Chen / Journal of Number Theory 111 ( Corollary 4.6. The formal power seres b(x and m(x are the unque solutons of the functonal equatons ( x b b(x = x (30 1 x and x x ( x m + x + ( x m = x, (31 1 x x 1 + x respectvely. The medan Bernoull numbers K n have certan connectons wth the number B n ( 1. Corollary 4.7. We have n+1 ( 1 n+1 K n = m=0 ( n ( 1 m m B m ( 1 m (3 and n B n ( 1 = m=0 ( n m+1 K m. (33 m Proof. Recall that t snh t = tet e t 1 = n B n ( 1 tn (n! for B n+1 ( 1 = 0. So, by applyng the formal Laplace transform, we get (x = n B n ( 1 xn+1. (34 Upon substtutng ths n Eq. (9, and replacng 4x /(1 x by y, weget ( y n+1 ( K n y n+1 = n B n ( y n 1 4 4

16 K.-W. Chen / Journal of Number Theory 111 ( = = y n+1 ( 1 n+1 n 1 y n+1 m=0 m=0 ( n ( 1 m m B m ( 1 m ( n ( 1 n+1+m m n 1 B m ( 1 m. Comparng the coeffcents of y n on the two sdes then yelds Eq. (3. Eq. (33 s followed mmedately usng the bnomal transform on Eq. (3. 5. Contnued fractons and Hankel determnants Rogers [11] found the contnued fracton expresson of (x as (x = x x 16x x = x 16x x x n 4 x... (35 + n Now usng Eq. (9 and replacng 4x /(1 x by y, we get the contnued fracton expanson for m(x. Proposton 5.1. The ordnary generatng functon m(x of the medan Bernoull numbers satsfes the contnued fracton representaton m(x = x x 16x x (x x (n 1 4 x + 4n 1 +. (36 (n 4 x (4n + 1(x Values of Hankel determnants of a sequence are known to be related to coeffcents of contnued fracton expansons of ts ordnary generatng functon. Proposton 5. (Krattenthaler [10, Theorem 11]. Let (μ n n 0 be a sequence of numbers wth generatng functon μ n t n+1 wrtten n the form μ n t n+1 = μ 0 t b 1 t b t.... ( a 0 t 1 + a 1 t 1 + a t Then the Hankel determnant det 0,j n (μ +j equals μ n+1 0 b n 1 bn 1 b n 1 b n.

17 388 K.-W. Chen / Journal of Number Theory 111 ( We need to change the representaton form of m(x n Proposton 5.1 as the form as Eq. (37. The followng two lemmas are needed. Lemma 5.3 (Dumont [4]. The followng representatons of a seres f(x are equvalent: f(x = x c 1 x c x c 3 x = x c 1 c x c 3 c 4 x 1 + c 1 x 1 + (c + c 3 x 1 + (c 4 + c 5 x.... Lemma 5.4. Let {b n } n 1 and {c n } n 1 be two sequences of complex numbers. If then, for n 1, we have 1 + θx + b 1x b x b 3 x b 4 x θx θx +... = 1 + c 1x c x c 3 x c 4 x..., ( c 1 = b 1 + θ, c 1 c = b 1 b, c n + c n+1 = b n + b n+1 + θ, c n+1 c n+ = b n+1 b n+. (39 Proof. Denote by A 0 (x and B 0 (x the left- and rght-hand sde of Eq. (38 and set and A 0 (x = 1 + θx + B 0 (x = 1 + b 1 x 1 + b x A (x c 1 x 1 + c. x B (x From the equalty A 0 (x = B 0 (x, we derve mmedately the followng: c 1 = b 1 + θ, c 1 c = b 1 b

18 K.-W. Chen / Journal of Number Theory 111 ( and A (x + b x = B (x + c x. The proof can then be readly completed by nducton. ( Now we can obtan the Hankel determnant det 0,j n K+j. Theorem 5.5. The Hankel determnant of the medan Bernoull numbers s ( ( det K+j = 1 n+1 n 0,j n =1 ( ( n +1 (4 3(4 1. (40 (4 + 1 Proof. We frst rewrte m(x as the form as the left-hand sde of Eq. (38 m(x = x b 1 x 1 + θx b x b 3 x b 4 x 1 + θx θx +..., where θ = 1 4 and for n 1, b n 1 = b n = Then from Lemmas 5.3 and 5.4 we have and for n 1, (n 1 4 4(4n 3(4n 1, (n 4 4(4n 1(4n + 1. m(x = x c 1 x c x c 3 x = x c 1 c x c 3 c 4 x 1 + c 1 x 1 + (c + c 3 x 1 + (c 4 + c 5 x... c 1 = b 1 + θ = 1 3, c 1 c = b 1 b = 1 40, c n + c n+1 = b n + b n+1 + θ = 8n4 + 8n 3 + 6n + n 1, (4n + 3(4n 1

19 390 K.-W. Chen / Journal of Number Theory 111 ( c n+1 c n+ = b n+1 b n+ = Applyng Proposton 5., we have ( det (K +j = 1 n+1 n (c 1 c n+1 0,j n =1 n+1 n ( = ( 1 =1 (n (n (4n + 1(4n + 3 (4n + 5. ( (4 3(4 1 (4 + 1 n +1. In vew of Eqs. (3 and (33, t seems that the Hankel determnant of B n ( 1 shall have the smlar formula. Corollary 5.6. det (B +j ( 1 0,j n = n ( ( n +1 (4 3(4 1. (41 (4 + 1 =1 Proof. Upon usng Eqs. (34 and (35, we have n B n ( 1 x(n+1 = x x 16x 3 4 x n 4 x n Thus n B n ( 1 xn+1 = x x 16x n 4 x n Now we use the smlar method as the proof of Theorem 5.5 and let 4x be y, then we get our concluson. Remark 5.7. It s worth notng that ( n+1 det (B +j ( 1 0,j n = det 0,j n (K +j. (4

20 Acknowledgments K.-W. Chen / Journal of Number Theory 111 ( The author would lke to thank the referee for some useful comments and suggestons. Ths research was supported by the Natonal Scence Foundaton of Tawan, Republc of Chna, Grant NSC M References [1] S. Akyama, Y. Tangawa, Multple zeta values at non-postve ntegers, Ramanujan J. 5 (4 ( [] K.-W. Chen, Applcatons related to the generalzed Sedel matrx, Ars Combn. (003 1pp, accepted for publcaton. [3] D. Dumont, Matrces d Euler-Sedel, n: Sém. Lothar. Combn., 5-éme Sesson, 1981, Publ. IRMA Strasbourg, vol. 18/s-04, 198, pp [4] D. Dumont, Further trangles of Sedel Arnold type and contnued fractons related to Euler and Sprnger numbers, Adv. Appl. Math. 16 (3 ( (do: /aama [5] D. Dumont, J. Zeng, Further results on the Euler and Genocch numbers, Aequatones Math. 47 ( [6] I.M. Gessel, Applcatons of the classcal umbra calculus, Algebra Unversals 49 ( [7] A. Granvlle, Arthmetc propertes of bnomal coeffcents, Canad. Math. Soc. Conf. Proc. 0 ( [8] K. Ireland, M. Rosen, A Classcal Introducton to Modern Number Theory, Sprnger, New York, 198. [9] M. Kaneko, A recurrence formula for the Bernoull numbers, Proc. Japan Acad. Ser. A Math. Sc. 71 ( [10] C. Krattenthaler, Advanced determnant calculus, Sém. Lothar. Combn. vol. 4 (The Andrews Festschrft, 1999, artcle B4q, 67pp. [11] L.J. Rogers, On the representaton of certan asymptotc seres as convergent contnued fractons, Proc. London Math. Soc. 4 ( (

Hyper-Sums of Powers of Integers and the Akiyama-Tanigawa Matrix

Hyper-Sums of Powers of Integers and the Akiyama-Tanigawa Matrix 6 Journal of Integer Sequences, Vol 8 (00), Artcle 0 Hyper-Sums of Powers of Integers and the Ayama-Tangawa Matrx Yoshnar Inaba Toba Senor Hgh School Nshujo, Mnam-u Kyoto 60-89 Japan nava@yoto-benejp Abstract

More information

Bernoulli Numbers and Polynomials

Bernoulli Numbers and Polynomials Bernoull Numbers and Polynomals T. Muthukumar tmk@tk.ac.n 17 Jun 2014 The sum of frst n natural numbers 1, 2, 3,..., n s n n(n + 1 S 1 (n := m = = n2 2 2 + n 2. Ths formula can be derved by notng that

More information

Determinants Containing Powers of Generalized Fibonacci Numbers

Determinants Containing Powers of Generalized Fibonacci Numbers 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 19 (2016), Artcle 1671 Determnants Contanng Powers of Generalzed Fbonacc Numbers Aram Tangboonduangjt and Thotsaporn Thanatpanonda Mahdol Unversty Internatonal

More information

Foundations of Arithmetic

Foundations of Arithmetic Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an

More information

Linear Algebra and its Applications

Linear Algebra and its Applications Lnear Algebra and ts Applcatons 4 (00) 5 56 Contents lsts avalable at ScenceDrect Lnear Algebra and ts Applcatons journal homepage: wwwelsevercom/locate/laa Notes on Hlbert and Cauchy matrces Mroslav Fedler

More information

arxiv: v1 [math.co] 12 Sep 2014

arxiv: v1 [math.co] 12 Sep 2014 arxv:1409.3707v1 [math.co] 12 Sep 2014 On the bnomal sums of Horadam sequence Nazmye Ylmaz and Necat Taskara Department of Mathematcs, Scence Faculty, Selcuk Unversty, 42075, Campus, Konya, Turkey March

More information

The Jacobsthal and Jacobsthal-Lucas Numbers via Square Roots of Matrices

The Jacobsthal and Jacobsthal-Lucas Numbers via Square Roots of Matrices Internatonal Mathematcal Forum, Vol 11, 2016, no 11, 513-520 HIKARI Ltd, wwwm-hkarcom http://dxdoorg/1012988/mf20166442 The Jacobsthal and Jacobsthal-Lucas Numbers va Square Roots of Matrces Saadet Arslan

More information

Binomial transforms of the modified k-fibonacci-like sequence

Binomial transforms of the modified k-fibonacci-like sequence Internatonal Journal of Mathematcs and Computer Scence, 14(2019, no. 1, 47 59 M CS Bnomal transforms of the modfed k-fbonacc-lke sequence Youngwoo Kwon Department of mathematcs Korea Unversty Seoul, Republc

More information

FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP

FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Linear Algebra APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of

More information

Beyond Zudilin s Conjectured q-analog of Schmidt s problem

Beyond Zudilin s Conjectured q-analog of Schmidt s problem Beyond Zudln s Conectured q-analog of Schmdt s problem Thotsaporn Ae Thanatpanonda thotsaporn@gmalcom Mathematcs Subect Classfcaton: 11B65 33B99 Abstract Usng the methodology of (rgorous expermental mathematcs

More information

The binomial transforms of the generalized (s, t )-Jacobsthal matrix sequence

The binomial transforms of the generalized (s, t )-Jacobsthal matrix sequence Int. J. Adv. Appl. Math. and Mech. 6(3 (2019 14 20 (ISSN: 2347-2529 Journal homepage: www.jaamm.com IJAAMM Internatonal Journal of Advances n Appled Mathematcs and Mechancs The bnomal transforms of the

More information

Restricted divisor sums

Restricted divisor sums ACTA ARITHMETICA 02 2002) Restrcted dvsor sums by Kevn A Broughan Hamlton) Introducton There s a body of work n the lterature on varous restrcted sums of the number of dvsors of an nteger functon ncludng

More information

Affine transformations and convexity

Affine transformations and convexity Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/

More information

Example: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,

Example: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41, The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson

More information

Self-complementing permutations of k-uniform hypergraphs

Self-complementing permutations of k-uniform hypergraphs Dscrete Mathematcs Theoretcal Computer Scence DMTCS vol. 11:1, 2009, 117 124 Self-complementng permutatons of k-unform hypergraphs Artur Szymańsk A. Paweł Wojda Faculty of Appled Mathematcs, AGH Unversty

More information

Randić Energy and Randić Estrada Index of a Graph

Randić Energy and Randić Estrada Index of a Graph EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL

More information

h-analogue of Fibonacci Numbers

h-analogue of Fibonacci Numbers h-analogue of Fbonacc Numbers arxv:090.0038v [math-ph 30 Sep 009 H.B. Benaoum Prnce Mohammad Unversty, Al-Khobar 395, Saud Araba Abstract In ths paper, we ntroduce the h-analogue of Fbonacc numbers for

More information

The lower and upper bounds on Perron root of nonnegative irreducible matrices

The lower and upper bounds on Perron root of nonnegative irreducible matrices Journal of Computatonal Appled Mathematcs 217 (2008) 259 267 wwwelsevercom/locate/cam The lower upper bounds on Perron root of nonnegatve rreducble matrces Guang-Xn Huang a,, Feng Yn b,keguo a a College

More information

Some congruences related to harmonic numbers and the terms of the second order sequences

Some congruences related to harmonic numbers and the terms of the second order sequences Mathematca Moravca Vol. 0: 06, 3 37 Some congruences related to harmonc numbers the terms of the second order sequences Neşe Ömür Sbel Koaral Abstract. In ths aer, wth hels of some combnatoral denttes,

More information

Perron Vectors of an Irreducible Nonnegative Interval Matrix

Perron Vectors of an Irreducible Nonnegative Interval Matrix Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of

More information

Evaluation of a family of binomial determinants

Evaluation of a family of binomial determinants Electronc Journal of Lnear Algebra Volume 30 Volume 30 2015 Artcle 22 2015 Evaluaton of a famly of bnomal determnants Charles Helou Pennsylvana State Unversty, cxh22@psuedu James A Sellers Pennsylvana

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

A property of the elementary symmetric functions

A property of the elementary symmetric functions Calcolo manuscrpt No. (wll be nserted by the edtor) A property of the elementary symmetrc functons A. Esnberg, G. Fedele Dp. Elettronca Informatca e Sstemstca, Unverstà degl Stud della Calabra, 87036,

More information

General viscosity iterative method for a sequence of quasi-nonexpansive mappings

General viscosity iterative method for a sequence of quasi-nonexpansive mappings Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 9 (2016), 5672 5682 Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence,

More information

J. Number Theory 130(2010), no. 4, SOME CURIOUS CONGRUENCES MODULO PRIMES

J. Number Theory 130(2010), no. 4, SOME CURIOUS CONGRUENCES MODULO PRIMES J. Number Theory 30(200, no. 4, 930 935. SOME CURIOUS CONGRUENCES MODULO PRIMES L-Lu Zhao and Zh-We Sun Department of Mathematcs, Nanjng Unversty Nanjng 20093, People s Republc of Chna zhaollu@gmal.com,

More information

Formulas for the Determinant

Formulas for the Determinant page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use

More information

On Finite Rank Perturbation of Diagonalizable Operators

On Finite Rank Perturbation of Diagonalizable Operators Functonal Analyss, Approxmaton and Computaton 6 (1) (2014), 49 53 Publshed by Faculty of Scences and Mathematcs, Unversty of Nš, Serba Avalable at: http://wwwpmfnacrs/faac On Fnte Rank Perturbaton of Dagonalzable

More information

A new Approach for Solving Linear Ordinary Differential Equations

A new Approach for Solving Linear Ordinary Differential Equations , ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of

More information

Ballot Paths Avoiding Depth Zero Patterns

Ballot Paths Avoiding Depth Zero Patterns Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,

More information

GELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n

GELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n GELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n KANG LU FINITE DIMENSIONAL REPRESENTATIONS OF gl n Let e j,, j =,, n denote the standard bass of the general lnear Le algebra gl n over the feld of

More information

Math 217 Fall 2013 Homework 2 Solutions

Math 217 Fall 2013 Homework 2 Solutions Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

Remarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence

Remarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence Remarks on the Propertes of a Quas-Fbonacc-lke Polynomal Sequence Brce Merwne LIU Brooklyn Ilan Wenschelbaum Wesleyan Unversty Abstract Consder the Quas-Fbonacc-lke Polynomal Sequence gven by F 0 = 1,

More information

Appendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis

Appendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems

More information

SL n (F ) Equals its Own Derived Group

SL n (F ) Equals its Own Derived Group Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu

More information

COMBINATORIAL IDENTITIES DERIVING FROM THE n-th POWER OF A 2 2 MATRIX

COMBINATORIAL IDENTITIES DERIVING FROM THE n-th POWER OF A 2 2 MATRIX COMBINATORIAL IDENTITIES DERIVING FROM THE n-th POWER OF A MATRIX J Mc Laughln 1 Mathematcs Department Trnty College 300 Summt Street, Hartford, CT 06106-3100 amesmclaughln@trncolledu Receved:, Accepted:,

More information

Convexity preserving interpolation by splines of arbitrary degree

Convexity preserving interpolation by splines of arbitrary degree Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete

More information

Problem Solving in Math (Math 43900) Fall 2013

Problem Solving in Math (Math 43900) Fall 2013 Problem Solvng n Math (Math 43900) Fall 2013 Week four (September 17) solutons Instructor: Davd Galvn 1. Let a and b be two nteger for whch a b s dvsble by 3. Prove that a 3 b 3 s dvsble by 9. Soluton:

More information

First day August 1, Problems and Solutions

First day August 1, Problems and Solutions FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve

More information

A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS

A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs

More information

18.781: Solution to Practice Questions for Final Exam

18.781: Solution to Practice Questions for Final Exam 18.781: Soluton to Practce Questons for Fnal Exam 1. Fnd three solutons n postve ntegers of x 6y = 1 by frst calculatng the contnued fracton expanson of 6. Soluton: We have 1 6=[, ] 6 6+ =[, ] 1 =[,, ]=[,,

More information

On the average number of divisors of the sum of digits of squares

On the average number of divisors of the sum of digits of squares Notes on Number heory and Dscrete Mathematcs Prnt ISSN 30 532, Onlne ISSN 2367 8275 Vol. 24, 208, No. 2, 40 46 DOI: 0.7546/nntdm.208.24.2.40-46 On the average number of dvsors of the sum of dgts of squares

More information

ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION

ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION

More information

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng

More information

Short running title: A generating function approach A GENERATING FUNCTION APPROACH TO COUNTING THEOREMS FOR SQUARE-FREE POLYNOMIALS AND MAXIMAL TORI

Short running title: A generating function approach A GENERATING FUNCTION APPROACH TO COUNTING THEOREMS FOR SQUARE-FREE POLYNOMIALS AND MAXIMAL TORI Short runnng ttle: A generatng functon approach A GENERATING FUNCTION APPROACH TO COUNTING THEOREMS FOR SQUARE-FREE POLYNOMIALS AND MAXIMAL TORI JASON FULMAN Abstract. A recent paper of Church, Ellenberg,

More information

2.3 Nilpotent endomorphisms

2.3 Nilpotent endomorphisms s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms

More information

Another converse of Jensen s inequality

Another converse of Jensen s inequality Another converse of Jensen s nequalty Slavko Smc Abstract. We gve the best possble global bounds for a form of dscrete Jensen s nequalty. By some examples ts frutfulness s shown. 1. Introducton Throughout

More information

BOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS

BOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all

More information

Applied Mathematics Letters

Applied Mathematics Letters Appled Matheatcs Letters 2 (2) 46 5 Contents lsts avalable at ScenceDrect Appled Matheatcs Letters journal hoepage: wwwelseverco/locate/al Calculaton of coeffcents of a cardnal B-splne Gradr V Mlovanovć

More information

The Ramanujan-Nagell Theorem: Understanding the Proof By Spencer De Chenne

The Ramanujan-Nagell Theorem: Understanding the Proof By Spencer De Chenne The Ramanujan-Nagell Theorem: Understandng the Proof By Spencer De Chenne 1 Introducton The Ramanujan-Nagell Theorem, frst proposed as a conjecture by Srnvasa Ramanujan n 1943 and later proven by Trygve

More information

Discrete Mathematics

Discrete Mathematics Dscrete Mathematcs 30 (00) 48 488 Contents lsts avalable at ScenceDrect Dscrete Mathematcs journal homepage: www.elsever.com/locate/dsc The number of C 3 -free vertces on 3-partte tournaments Ana Paulna

More information

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not

More information

Lecture 5 Decoding Binary BCH Codes

Lecture 5 Decoding Binary BCH Codes Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture

More information

REGULAR POSITIVE TERNARY QUADRATIC FORMS. 1. Introduction

REGULAR POSITIVE TERNARY QUADRATIC FORMS. 1. Introduction REGULAR POSITIVE TERNARY QUADRATIC FORMS BYEONG-KWEON OH Abstract. A postve defnte quadratc form f s sad to be regular f t globally represents all ntegers that are represented by the genus of f. In 997

More information

A New Refinement of Jacobi Method for Solution of Linear System Equations AX=b

A New Refinement of Jacobi Method for Solution of Linear System Equations AX=b Int J Contemp Math Scences, Vol 3, 28, no 17, 819-827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,

More information

arxiv: v6 [math.nt] 23 Aug 2016

arxiv: v6 [math.nt] 23 Aug 2016 A NOTE ON ODD PERFECT NUMBERS JOSE ARNALDO B. DRIS AND FLORIAN LUCA arxv:03.437v6 [math.nt] 23 Aug 206 Abstract. In ths note, we show that f N s an odd perfect number and q α s some prme power exactly

More information

Zhi-Wei Sun (Nanjing)

Zhi-Wei Sun (Nanjing) Acta Arth. 1262007, no. 4, 387 398. COMBINATORIAL CONGRUENCES AND STIRLING NUMBERS Zh-We Sun Nanng Abstract. In ths paper we obtan some sophstcated combnatoral congruences nvolvng bnomal coeffcents and

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

1 Generating functions, continued

1 Generating functions, continued Generatng functons, contnued. Exponental generatng functons and set-parttons At ths pont, we ve come up wth good generatng-functon dscussons based on 3 of the 4 rows of our twelvefold way. Wll our nteger-partton

More information

Fixed points of IA-endomorphisms of a free metabelian Lie algebra

Fixed points of IA-endomorphisms of a free metabelian Lie algebra Proc. Indan Acad. Sc. (Math. Sc.) Vol. 121, No. 4, November 2011, pp. 405 416. c Indan Academy of Scences Fxed ponts of IA-endomorphsms of a free metabelan Le algebra NAIME EKICI 1 and DEMET PARLAK SÖNMEZ

More information

A FORMULA FOR COMPUTING INTEGER POWERS FOR ONE TYPE OF TRIDIAGONAL MATRIX

A FORMULA FOR COMPUTING INTEGER POWERS FOR ONE TYPE OF TRIDIAGONAL MATRIX Hacettepe Journal of Mathematcs and Statstcs Volume 393 0 35 33 FORMUL FOR COMPUTING INTEGER POWERS FOR ONE TYPE OF TRIDIGONL MTRIX H Kıyak I Gürses F Yılmaz and D Bozkurt Receved :08 :009 : ccepted 5

More information

The Order Relation and Trace Inequalities for. Hermitian Operators

The Order Relation and Trace Inequalities for. Hermitian Operators Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

The internal structure of natural numbers and one method for the definition of large prime numbers

The internal structure of natural numbers and one method for the definition of large prime numbers The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract

More information

MEM 255 Introduction to Control Systems Review: Basics of Linear Algebra

MEM 255 Introduction to Control Systems Review: Basics of Linear Algebra MEM 255 Introducton to Control Systems Revew: Bascs of Lnear Algebra Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty Outlne Vectors Matrces MATLAB Advanced Topcs Vectors A

More information

1 Matrix representations of canonical matrices

1 Matrix representations of canonical matrices 1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:

More information

Solutions Homework 4 March 5, 2018

Solutions Homework 4 March 5, 2018 1 Solutons Homework 4 March 5, 018 Soluton to Exercse 5.1.8: Let a IR be a translaton and c > 0 be a re-scalng. ˆb1 (cx + a) cx n + a (cx 1 + a) c x n x 1 cˆb 1 (x), whch shows ˆb 1 s locaton nvarant and

More information

ON THE NUMBER OF PRIMITIVE PYTHAGOREAN QUINTUPLES

ON THE NUMBER OF PRIMITIVE PYTHAGOREAN QUINTUPLES Journal of Algebra, Nuber Theory: Advances and Applcatons Volue 3, Nuber, 05, Pages 3-8 ON THE NUMBER OF PRIMITIVE PYTHAGOREAN QUINTUPLES Feldstrasse 45 CH-8004, Zürch Swtzerland e-al: whurlann@bluewn.ch

More information

a b a In case b 0, a being divisible by b is the same as to say that

a b a In case b 0, a being divisible by b is the same as to say that Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :

More information

Combinatorial Identities for Incomplete Tribonacci Polynomials

Combinatorial Identities for Incomplete Tribonacci Polynomials Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015, pp. 40 49 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM Combnatoral Identtes for Incomplete

More information

A combinatorial proof of multiple angle formulas involving Fibonacci and Lucas numbers

A combinatorial proof of multiple angle formulas involving Fibonacci and Lucas numbers Notes on Number Theory and Dscrete Mathematcs ISSN 1310 5132 Vol. 20, 2014, No. 5, 35 39 A combnatoral proof of multple angle formulas nvolvng Fbonacc and Lucas numbers Fernando Córes 1 and Dego Marques

More information

MULTIPLICATIVE FUNCTIONS: A REWRITE OF ANDREWS CHAPTER 6

MULTIPLICATIVE FUNCTIONS: A REWRITE OF ANDREWS CHAPTER 6 MULTIPLICATIVE FUNCTIONS: A REWRITE OF ANDREWS CHAPTER 6 In these notes we offer a rewrte of Andrews Chapter 6. Our am s to replace some of the messer arguments n Andrews. To acheve ths, we need to change

More information

Chowla s Problem on the Non-Vanishing of Certain Infinite Series and Related Questions

Chowla s Problem on the Non-Vanishing of Certain Infinite Series and Related Questions Proc. Int. Conf. Number Theory and Dscrete Geometry No. 4, 2007, pp. 7 79. Chowla s Problem on the Non-Vanshng of Certan Infnte Seres and Related Questons N. Saradha School of Mathematcs, Tata Insttute

More information

Discrete Mathematics. Laplacian spectral characterization of some graphs obtained by product operation

Discrete Mathematics. Laplacian spectral characterization of some graphs obtained by product operation Dscrete Mathematcs 31 (01) 1591 1595 Contents lsts avalable at ScVerse ScenceDrect Dscrete Mathematcs journal homepage: www.elsever.com/locate/dsc Laplacan spectral characterzaton of some graphs obtaned

More information

Complex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1)

Complex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1) Complex Numbers If you have not yet encountered complex numbers, you wll soon do so n the process of solvng quadratc equatons. The general quadratc equaton Ax + Bx + C 0 has solutons x B + B 4AC A For

More information

A Witt type formula. G.A.T.F.da Costa 1 Departamento de Matemática Universidade Federal de Santa Catarina Florianópolis-SC-Brasil.

A Witt type formula. G.A.T.F.da Costa 1 Departamento de Matemática Universidade Federal de Santa Catarina Florianópolis-SC-Brasil. A Wtt type formula G.A.T.F.da Costa 1 Departamento de Matemátca Unversdade Federal de Santa Catarna 88040-900-Floranópols-SC-Brasl arxv:1302.6950v2 [math.co] 4 Mar 2013 Abstract Gven a fnte, connected

More information

Bezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0

Bezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0 Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the

More information

A Simple Research of Divisor Graphs

A Simple Research of Divisor Graphs The 29th Workshop on Combnatoral Mathematcs and Computaton Theory A Smple Research o Dvsor Graphs Yu-png Tsao General Educaton Center Chna Unversty o Technology Tape Tawan yp-tsao@cuteedutw Tape Tawan

More information

An (almost) unbiased estimator for the S-Gini index

An (almost) unbiased estimator for the S-Gini index An (almost unbased estmator for the S-Gn ndex Thomas Demuynck February 25, 2009 Abstract Ths note provdes an unbased estmator for the absolute S-Gn and an almost unbased estmator for the relatve S-Gn for

More information

Amusing Properties of Odd Numbers Derived From Valuated Binary Tree

Amusing Properties of Odd Numbers Derived From Valuated Binary Tree IOSR Journal of Mathematcs (IOSR-JM) e-iss: 78-578, p-iss: 19-765X. Volume 1, Issue 6 Ver. V (ov. - Dec.016), PP 5-57 www.osrjournals.org Amusng Propertes of Odd umbers Derved From Valuated Bnary Tree

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0 MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

DIFFERENTIAL FORMS BRIAN OSSERMAN

DIFFERENTIAL FORMS BRIAN OSSERMAN DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne

More information

Smarandache-Zero Divisors in Group Rings

Smarandache-Zero Divisors in Group Rings Smarandache-Zero Dvsors n Group Rngs W.B. Vasantha and Moon K. Chetry Department of Mathematcs I.I.T Madras, Chenna The study of zero-dvsors n group rngs had become nterestng problem snce 1940 wth the

More information

Journal of Number Theory. On Euler numbers, polynomials and related p-adic integrals

Journal of Number Theory. On Euler numbers, polynomials and related p-adic integrals Journal of Number Theory 19 009 166 179 Contents lsts avalable at ScenceDrect Journal of Number Theory www.elsever.com/locate/jnt On Euler numbers, polynomals and related p-adc ntegrals Mn-Soo Km Natonal

More information

Some basic inequalities. Definition. Let V be a vector space over the complex numbers. An inner product is given by a function, V V C

Some basic inequalities. Definition. Let V be a vector space over the complex numbers. An inner product is given by a function, V V C Some basc nequaltes Defnton. Let V be a vector space over the complex numbers. An nner product s gven by a functon, V V C (x, y) x, y satsfyng the followng propertes (for all x V, y V and c C) (1) x +

More information

ON A DIOPHANTINE EQUATION ON TRIANGULAR NUMBERS

ON A DIOPHANTINE EQUATION ON TRIANGULAR NUMBERS Mskolc Mathematcal Notes HU e-issn 787-43 Vol. 8 (7), No., pp. 779 786 DOI:.854/MMN.7.536 ON A DIOPHANTINE EUATION ON TRIANGULAR NUMBERS ABDELKADER HAMTAT AND DJILALI BEHLOUL Receved 6 February, 5 Abstract.

More information

On the spectral norm of r-circulant matrices with the Pell and Pell-Lucas numbers

On the spectral norm of r-circulant matrices with the Pell and Pell-Lucas numbers Türkmen and Gökbaş Journal of Inequaltes and Applcatons (06) 06:65 DOI 086/s3660-06-0997-0 R E S E A R C H Open Access On the spectral norm of r-crculant matrces wth the Pell and Pell-Lucas numbers Ramazan

More information

Linear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.

Linear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space. Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +

More information

Lecture Notes Introduction to Cluster Algebra

Lecture Notes Introduction to Cluster Algebra Lecture Notes Introducton to Cluster Algebra Ivan C.H. Ip Updated: Ma 7, 2017 3 Defnton and Examples of Cluster algebra 3.1 Quvers We frst revst the noton of a quver. Defnton 3.1. A quver s a fnte orented

More information

The KMO Method for Solving Non-homogenous, m th Order Differential Equations

The KMO Method for Solving Non-homogenous, m th Order Differential Equations The KMO Method for Solvng Non-homogenous, m th Order Dfferental Equatons Davd Krohn Danel Marño-Johnson John Paul Ouyang March 14, 2013 Abstract Ths paper shows a smple tabular procedure for fndng the

More information

On quasiperfect numbers

On quasiperfect numbers Notes on Number Theory and Dscrete Mathematcs Prnt ISSN 1310 5132, Onlne ISSN 2367 8275 Vol. 23, 2017, No. 3, 73 78 On quasperfect numbers V. Sva Rama Prasad 1 and C. Suntha 2 1 Nalla Malla Reddy Engneerng

More information

DISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization

DISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization DISCRIMINANTS AND RAMIFIED PRIMES KEITH CONRAD 1. Introducton A prme number p s sad to be ramfed n a number feld K f the prme deal factorzaton (1.1) (p) = po K = p e 1 1 peg g has some e greater than 1.

More information

arxiv: v1 [math.co] 1 Mar 2014

arxiv: v1 [math.co] 1 Mar 2014 Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest

More information

Valuated Binary Tree: A New Approach in Study of Integers

Valuated Binary Tree: A New Approach in Study of Integers Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR) Volume 4, Issue 3, March 6, PP 63-67 ISS 347-37X (Prnt) & ISS 347-34 (Onlne) wwwarcournalsorg Valuated Bnary Tree: A ew Approach

More information

Characterizing the properties of specific binomial coefficients in congruence relations

Characterizing the properties of specific binomial coefficients in congruence relations Eastern Mchgan Unversty DgtalCommons@EMU Master's Theses and Doctoral Dssertatons Master's Theses, and Doctoral Dssertatons, and Graduate Capstone Projects 7-15-2015 Characterzng the propertes of specfc

More information

A Hybrid Variational Iteration Method for Blasius Equation

A Hybrid Variational Iteration Method for Blasius Equation Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method

More information

An efficient algorithm for multivariate Maclaurin Newton transformation

An efficient algorithm for multivariate Maclaurin Newton transformation Annales UMCS Informatca AI VIII, 2 2008) 5 14 DOI: 10.2478/v10065-008-0020-6 An effcent algorthm for multvarate Maclaurn Newton transformaton Joanna Kapusta Insttute of Mathematcs and Computer Scence,

More information

FINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN

FINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There

More information