On the transcendence of infinite sums of values of rational functions
|
|
- Janis Potter
- 5 years ago
- Views:
Transcription
1 O the trascedece of ifiite sus of values of ratioal fuctios N. Saradha ad R. Tijdea Abstract P () = We ivestigate coverget sus T = Q() ad U = P (X), Q(X) Q[X], ad Q(X) has oly siple ratioal roots. = ( ) P () Q() where Adhikari, Shorey ad the authors have show that T ad U are either ratioal or trascedetal. I the preset paper siple ecessary ad sufficiet coditios are forulated for the trascedece of T ad U if the degree of Q is 3 ad 2, respectively. Itroductio. Throughout the paper we take > to be a iteger ad f(x) a uber theoretic fuctio which is periodic od with f(i) = 0. We deote by Z, Q, Q, the rig of ratioal i= itegers, the field of ratioals ad the field of algebraic ubers. The first geeral result o the o-vaishig of S = = is probably due to Dirichlet who showed i 839 by his class uber forulas that L(, χ) = = χ() f() 0 if χ is a o-pricipal Dirichlet character. Aroud 970, Chowla ad Siegel obtaied soe o-vaishig results o periodic fuctios f assuig oly values, ad 0, see [4]. Chowla [4] ad Erdős (see [5]) forulated soe related cojectures, oe of which was proved by Baker, Birch ad Wirsig [3] i 973. They used Baker s theory o liear fors i logariths to establish that S 0 if f is a o-vaishig fuctio defied o the itegers with algebraic values ad period such that (i) f(r) = 0 if < gcd(r, ) <. (ii) the cyclotoic polyoial Φ is irreducible over Q(f(),..., f()).
2 If is prie, the (i) is vacuous ad if f is ratioal valued the (ii) holds trivially. Baker, Birch ad Wirsig further showed that their result would be false if (i) or (ii) is oitted. Ideed, let = p 2 where p is a prie ad f be defied by = f() = ( p s ) 2 ζ(s) where ζ(s) deotes the Riea zeta fuctio. For p = 2, this yields = 0 with period 4. Thus (i) is ecessary. To see that (ii) caot be oitted, they cosidered the uadratic characters χ, χ od 2 with coductors 3 ad 4, respectively, ad f = 2χ 3χ. The vaishig of S follows iediately fro the values L(, χ) = π 2 ad L(, 3 χ ) = π. They 3 also characterised all odd algebraic valued fuctios f for which S = 0. I 982 Okada [6] published a result which provides a descriptio of all fuctios for which (ii) holds ad S = 0. The criterio is a syste of ϕ() + ω() hoogeeous liear euatios i f(),..., f() with ratioal coefficiets where ϕ() deotes the Euler s totiet fuctio ad ω() is the uber of distict prie divisors of. The precise result is stated i Sectio 2. Okada s proof depeds o the basic result o the liear idepedece of the logariths of algebraic ubers ad o the o-vaishig of L(, χ) if χ is a o-pricipal Dirichlet character. Okada s result was used by Tijdea [7] to prove that S 0 if f : Z Q is copletely ultiplicative, ad if f is ultiplicative such that f(p k ) < p for every prie divisor p of ad every positive iteger k (see Sectio 2). I 200, Adhikari, Saradha, Shorey ad Tijdea [] proved that if S 0, the S is trascedetal. They used this result to prove that if P (X) Q[X] ad Q(X) Q[X] where Q(X) is a polyoial with siple ratioal roots, the T = = P () Q() is trascedetal provided T coverges ad is irratioal. They proved ay related results. Thus the uestio of decidig whether S = 0 or ot has gaied iportace. iforatio o the developets sketched above we refer to [2] ad [7]. For ore I the preset paper, we rephrase Okada s theore so that it becoes a decopositio f(a) lea (Lea ) ad use it to derive that S = 0 iplies = 0 for ay positive = iteger a coprie to (Corollary ). This is a iterediate result i [3, p.23, forula (7)]. 2
3 Lea ad Corollary are used i Sectio 3 to prove that ( ) (a + b) () ( + s )( + s 2 ) with a, b, s, s 2 Z, s s 2, a + b > 0 ad s, s 2 ever a o-egative iteger, is trascedetal except whe s s 2 (od ) ad a = 0. I the exceptioal case the su is ratioal. I Sectio 4, we use Lea to prove that a + b (2) ( + s )( + s 2 )( + s 3 ) with a, b, s, s 2, s 3 Z, s, s 2, s 3 distict, a 2 + b 2 0 ad s /, s 2 /, s 3 / ever a oegative iteger, is trascedetal except whe s s 2 s 3 (od ) or s s 2 (od ) ad as 3 = b or s s 3 (od ) ad as 2 = b or s 2 s 3 (od ) ad as = b. I the exceptioal cases the su is ratioal. O the other had, the exaples ( ) (3 + )(3 + 3)(3 + 5) = 6 ; (4 + )(4 + 2)(4 + 3)(4 + 4) = 0 show that the correspodig results for () ad (2) whe the deoiators are replaced with 3 ad 4 factors, respectively, are ot valid. 2 Decopositio Lea. We first itroduce soe otatio ad state Okada s ai result. We deote by P the set of all pries dividig. For p P ad Z, we deote by v p () the expoet to which p divides. We write J = {a Z a, gcd (a, ) = }, ad L = {r Z r, < gcd (r, ) < } L = L {}. We defie for r L ad p P, P (r) = {p P v p (r) v p ()} ad v p () + if p P (r) p ε(r, p) = v p (r) otherwise. 3
4 We further defie for r L ad a J, where ad Theore A. if A(r, a) = gcd(r, ) S(r) = p P (r) p P (r) ( p ϕ() ) S(r) p α(p) 0 α(p) < ϕ() if r a gcd(r, ) (od ) δ(r, a, ) = 0 otherwise δ(r, a, ) (Okada.) If Φ is irreducible over Q(f(),..., f()), the S = 0 if ad oly (3) f(a) + r L f(r)a(r, a) + f() ϕ() = 0 for a J ad (4) r L f(r)ε(r, p) = 0 for p P. We observe that (3) ad (4) for a syste of ϕ()+ω() hoogeeous liear euatios i f(),..., f() with o-egative ratioal coefficiets. Suppose f 0 ad S = 0. By (4), f() = 0 if f(r) = 0 for each r L. Hece, by (3), there exists at least oe r L for which f(r) 0. So Theore A iplies the result of Baker, Birch ad Wirsig etioed i Sectio. I particular we fid that if is prie, the S 0 i accordace with Chowla s cojecture. Now we give a euivalet versio of Theore A. Lea. (Decopositio lea.) Let Φ be irreducible over Q(f(),..., f()). Let M be the set of positive itegers which are coposed of prie factors of ad let v p () + ε(r, p) = if v p p(r) v p () v p (r) otherwise 4
5 The S = 0 if ad oly if (5) M f(a) = 0 for every a with 0 < a <, gcd(a, ) = ad (6) r= gcd(r,)> f(r)ε(r, p) = 0 for every prie divisor p of. Proof. show that Note that ε(r, p) 0 iplies p r, hece r L. So, by Theore A, it suffices to (7) f(a) + r L f(r)a(r, a) + f() ϕ() = f(a). M For ay iteger r, we deote by M(r) the set of positive itegers which are coposed of prie factors fro P (r). Thus M(r) = M(gcd(r, )) ad M = M(). We cosider f(a) f(a) f(a) = f(a) + M r L M a rod = f(a) + f(r) gcd(r, ) r L M(r) δ(r, a, ) + M + f() M() =: V. (8) We have M() = ( + p + p + ) = 2 p p p = ϕ(). Let M(r) ad δ(r, a, ) =. We have v p () = 0 if p P (r). If p P (r), the gcd(p, gcd(r,) ) = whece pϕ( gcd(r,) ) (od gcd(r,) ). Sice ϕ( gcd(r,) p ϕ() gcd(r, ) gcd(r, )(od ) whece δ(r, a, p ϕ() ) = δ(r, a, ). Thus (9) M(r) δ(r, a, ) = S(r) δ(r, a, ) p P (r) ( + p + + ). ϕ() p2ϕ() ) ϕ(), we obtai We substitute (8) ad (9) i the expressio V to obtai (7). [3]. As a coseuece of Lea we derive the followig corollary which is forula (7) of 5
6 Corollary. Let Φ be irreducible over Q(f(),, f()). Suppose S = 0. The = f(k) = 0 for every k with gcd(k, ) =. Proof. By Lea, we fid that (5) ad (6) hold. Thus, by (5), M f(ak) = M f(a ) = 0 for every a with a <, gcd(a, ) = where a ak (od ). If kr r (od ), the ε(r, p) = ε(r, p). Hece r L f(kr)ε(r, p) = r L f(r )ε(r, p) = 0 by (6). Now the assertio follows fro the coverse part of Lea. We write forally = f() = gcd(a,)= a M() f(a). It follows fro Lea that if the series o the left had side vaishes, so does the expressio withi brackets for every a coprie to. The coverse is ot true i geeral. For istace, if = 2 ad f() = ( ), Z, the for every odd a, but M(2) f(a) = f() f(2a) = f(a) + + f(4a) + = = ( ) = log 2 0. = I this exaple we have r L f(r)ε(r, p) = 2f(2) 0 ad hece (6) is ot satisfied. As aother coseuece of Lea, we derive Theores 9 ad 0 of [7]. Corollary 2. Let f be copletely ultiplicative, or ultiplicative with f(p k ) < p for every prie divisor p of ad every positive iteger k. Also let Φ be irreducible over Q(f(),..., f()). The S 0. 6
7 Proof. Suppose S = 0. Sice f is ultiplicative, (5) with a = iplies that M f() = 0. Note that the series is absolutely coverget because of the periodicity of f. Sice f() = ( ) f(p j ), p j M p j=0 we get (0). j=0 f(p j ) p j = 0 for soe prie divisor p of If f is copletely ultiplicative, the ( ) j f(p) = 0 for soe prie divisor p of j=0 p which is ot possible. If f is ultiplicative with f(p k ) < p for every prie divisor p of ad every positive iteger k, the j=0 f(p j ) p j > p p ( + p + ) = 0 cotradictig (0). We reark here that i the stateet of Theore 9 i [7], the coditio f(p k ) p should be replaced by f(p k ) < p. The exaple f(0) = 2, f() = f(5) =, f(2) = f(4) =, f(3) = 2 with period 6 shows that the stateet is false uder the forer coditio. The applicatio to Erdős proble i [7] is ot affected by this correctio. 3 Alteratig Series. I this sectio, we apply Lea ad Corollary to ivestigate sus () T = ( ) (α + β) ( + s )( + s 2 ) with α, β Q, s, s 2 Z. We prove 7
8 Theore. Suppose T is give by () with α + β > 0. Let Φ 2 be irreducible over Q(α, β) ad s, s 2 distict itegers such that +s, +s 2 do ot vaish for 0. Assue that α 0 if s s 2 (od ). The T is trascedetal. We derive Theore fro the followig lea. Lea 2. Let T = ( ) (α + β ) ( + r )( + r 2 ) with 0 < r, 0 < r 2, r r 2, r, r 2 Z, α, β Q. Suppose α + β > 0 ad Φ 2 is irreducible over Q(α, β ). The T is trascedetal. Proof. First we show that T 0. Suppose T = 0. We ay assue without loss of geerality that gcd(, r, r 2 ) =. By partial fractios, we see that T = { } A ( ) + B + r + r 2 where A = β α r (r r 2 ) ad B = β α r 2 (r r 2 ). Suppose A = 0. The B 0 ad T = B { }. 2 + r r 2 Hece T 0. Siilarly if A 0 ad B = 0, we have T 0. Thus we ay suppose that A 0, B 0. We defie f() for 0 as A if r (od 2) B if r 2 (od 2) f() = A if + r (od 2) B if + r 2 (od 2) 0 otherwise. Thus f is a periodic fuctio with period 2 ad (2) T = = f() = { f(r ) + f(r } 2) f(r ) f(r 2 ) = r 2 + r r r 2 Hece (5) ad (6) are valid by Lea. 8
9 Case. Let be odd. The oe of r i, + r i is eve ad the other is odd for i =, 2. Further ε(r i, 2) = 0 if r i is odd ad ε(r i, 2) = 2 if r i is eve. We apply (6) for p = 2 to obtai f(r ) = f(r 2 ). We re-write (2) as T = f(r ) { (2 + r )(2 + + r ) ± (2 + r 2 )(2 + + r 2 ) We see that all ters withi the curly brackets have the sae sig whece T 0, which is a cotradictio. Thus is ot odd. }. Case 2. Let be eve. The r i, + r i are both odd or both eve for i =, 2. By (5) ad (6), there exists a r L for which f(r) 0. Let r = r ad p be a prie factor of dividig r. The it follows fro (6) that there exists s 2 with p s ad s r such that f(s) 0. Now s r 2, s + r 2, sice gcd(, r, r 2 ) =. Hece s = + r. Assue gcd(r 2, 2) =. The gcd( +r 2, 2) =. Suppose v 2 (r ) v 2 (). The either v 2 (r ) > v 2 (), v 2 ( + r ) = v 2 () or v 2 (r ) = v 2 (), v 2 ( + r ) > v 2 (). O applyig (6) with p = 2, we obtai ε(r, 2) = ε( + r, 2). Sice v 2 (r ) v 2 ( + r ), this is possible oly whe v 2 (r ) v 2 (2) ad v 2 ( + r ) v 2 (2) which is false. Thus we get (3) v 2 (r ) = v 2 ( + r ) < v 2 (). Next we show that r 2, r 2 + p,, r 2 + ( p ) p are all coprie to 2. Suppose there exists a prie p dividig both r 2 + i p ad 2 for soe i with 0 < i < p. The p = p, v p () = ad there exists exactly oe such i, say i 0. Now we apply (5) with a = r 2 ad r 2 + i to see that p f(r 2 ) = f(r 2 + i p ) for 0 < i < p, i i 0, sice all the ters i (5) for > are eual because f(a) 0 for >, M oly whe a r or + r (od 2) ad p divides r ad. Sice r 2 ad + r 2 are the oly itegers k od 2 coprie to with f(k) 0, we coclude that p = 2. Thus r = R 2 α, + r = R 2 2 α with gcd(r, 2) = gcd(r 2, 2) = ad 0 < α < v 2 () by (3). We apply (5) with a = R, R 2 to get f(r ) + f(r 2 α ) 2 α = 0, f(r 2 ) + f(r 22 α ) 2 α = 0. 9
10 Hece f(r ) 0, f(r 2 ) 0. Thus {R, R 2 } = {r 2, + r 2 } which gives = R 2 R iplyig α = 0, which is a cotradictio. This proves that gcd(r i, 2) > for i =, 2. Note that r or r 2 is odd. Assue r is odd. (The case r 2 is odd is siilar.) Put d = gcd(r, 2) >. Hece d is odd. We put a = r d, b = +r d Now we show that it is possible to choose a iteger such that. The gcd(a, 2 (4) is prie, > 2, + (od 2), a b (od 2 d ). d ) = gcd(b, 2 d ) =. If a b + (od 2 ), the we choose ay prie > 2 which is d a b (od 2 ) ad (4) d is satisfied. So we suppose that a b + (od 2 2 ). There are ϕ(2)/ϕ( ) priitive d d residue classes od 2 i the residue class a b (od 2 ). Now d ϕ(2) ϕ( 2 ) d ( p ) 2 d p d p prie sice d is odd. Hece we ay take a priitive residue class + (od 2) ad a b (od 2 ). By Dirichlet s theore, we ca fid a prie satisfyig (4). Note that d (5) (r + ) = ad + bd + = r + + r (od 2). Sice gcd(, 2) =, we ay apply Corollary with k = to obtai { f(r ) (6) + f(r 2) f(r ) f(r } 2) = r 2 + r2 2 + r3 2 + r4 where r i r i (od 2), 0 < r i 2 for i 4 with r 3 = + r, r 4 = + r 2. We have (7) r 3 r 3 = ( + r ) ab bd = ad = r (od 2) ad by (5), (8) r r (( + r )) r 3 (od 2). Thus r 3 = r, r = r 3 = + r. Fro (5), it follows that ( )r 0 (od ). Suppose r 2 r 2 (od ). The ( )r 2 0 (od ). Hece (od ) sice gcd(, r, r 2 ) =. If (od 2), the a b(od 2 ) which is ot possible sice b a =. O the other d d had, + (od 2) by (4). Thus r2 r 2 (od ). Now we add (2) with (6) ad use (7), (8) to get { ( ) ( f(r 2 ) r r r2 0 ) } = r4
11 Note that r 4 r 4 = ( + r 2 ) + r 2 (od 2) so that r 4 = r 2 ±. By the ootoicity of with respect to k, the sig of the expressio betwee the 2+k 2+k+ curly brackets i the ifiite su above depeds oly o r 2 ad r2. Hece all these expressios have the sae sig. Sice r 2 r 2 (od ), the su does ot vaish. Hece f(r 2 ) = 0, which is a cotradictio. Thus T 0. Now we apply [, Theore ] to see that T is trascedetal. Proof of Theore. usig partial fractios, we have We assue without loss of geerality that gcd(, s, s 2 ) =. By T = { A ( ) + B } + s + s 2 where A = β αs (s s 2 ) ad B = β αs 2 (s s 2 ). Let s r (od ) ad s 2 r 2 (od ) with 0 < r, 0 < r 2. The (9) T = γ ± { A ( ) + B } + r + r 2 where γ Q(α, β). If r = r 2, the (20) T = γ ± α ( ) + r = γ ± α { }. 2 + r r We see that the ifiite su i the latter expressio for T does ot vaish. Now we apply [, Theore ] with if r (od 2) f() = if + r (od 2) 0 otherwise to coclude that the ifiite series ad hece T is trascedetal. Thus we ay assue that r r 2. Now we apply Lea 2 to (9) to see that T is trascedetal. We ote that if s s 2 (od ) ad α = 0, the by (20), we have T algebraic.
12 4 Series with ters havig three products i the deoiator. I this sectio we cosider series of the for α + β (2) U = ( + s )( + s 2 )( + s 3 ) with α, β Q, s, s 2, s 3 Z. We prove Theore 2. Suppose U is give by (2) with α + β > 0. Let Φ be irreducible over Q(α, β) ad s, s 2, s 3 be distict itegers such that + s, + s 2, + s 3 do ot vaish for 0. Assue that s, s 2, s 3 are ot i the sae residue class od. Further let (22) s s 2 (od ) if αs 3 = β; s 2 s 3 (od ) if αs = β; s 3 s (od ) if αs 2 = β. The U is trascedetal. We derive Theore 2 fro the followig lea. Lea 3. Let U = α + β ( + r )( + r 2 )( + r 3 ) with r, r 2, r 3 distict positive itegers ad α, β is irreducible over Q(α, β ). The U is trascedetal. Q. Suppose α + β > 0 ad Φ Proof. First we show that U 0. Suppose U = 0. We ay assue without loss of geerality that gcd(, r, r 2, r 3 ) =. Usig partial fractios, we get where A = U = { A β α r (r r 2 )(r r 3 ), B = + r + B + r 2 + C + r 3 β α r 2 (r 2 r )(r 2 r 3 ), C = } β α r 3 (r 3 r )(r 3 r 2 ). Observe that A + B + C = 0. Hece if ay of A, B, C vaishes the the series U reduces to soe series i which all the expressios betwee curly brackets have the sae sig. The U 0. Thus we ay assue that oe of A, B, C vaishes. We defie f() for 0 as 2
13 A if r (od ) B if r 2 (od ) f() = C if r 3 (od ) 0 otherwise. Thus f is a periodic fuctio with period takig oly three o-zero values f(r ), f(r 2 ), f(r 3 ) with (23) f(r ) + f(r 2 ) + f(r 3 ) = 0 ad U = = f() = 0. Hece (5) ad (6) are valid by Lea. Thus there exist r, s {r, r 2, r 3 } with r s ad a prie p dividig gcd(, r, s). Without loss of geerality, we ay take r = r, s = r 2. If p, by (6) with p = p, we have f(r ) + f(r 2 ) = 0 which gives f(r 3 ) = 0 by (23), which is a cotradictio. Thus p 2. Now suppose gcd(r 3, ) =. Let r 3 + i p a i (od ) where 0 < a i < for 0 i < p. We ote that a i s are all distict ad coprie to. Whe we apply (5) to the ubers a i, the all o-zero ters correspodig to > are the sae ad we fid f(a i ) = f(r 3 ) 0 for i = 0,,..., p. Sice p 2, this is a cotradictio. Thus gcd(r 3, ) >. Further we ay suppose that p r 3 sice gcd(, r, r 2, r 3 ) =. Let be a prie with gcd(r 3, ). By applyig (6) with p =, we ay assue that r 2. The r. Thus we have r = R p α, r 2 = R 2 p β γ, r 3 = R 3 δ where the dots represet other prie factors of ad gcd(r i, ) = for i =, 2, 3. Applyig (6) with p = p ad p =, we fid that f(r ) ad f(r 2) f(r 2 ) f(r 3 are egative ad, by (23), that ) f(r 2 ) = f(r ) + f(r 3 ). Hece f(r 2 ) > f(r ) ad f(r 2 ) > f(r 3 ). Agai usig (6) we get α > β, δ > γ ad v p (r 2 ) < v p (), v (r 2 ) < v (). It follows that gcd(r, r 3, ) = ad that v p (r 2 ) < v p () for every prie divisor p of. Hece we ay write r = R p α p αt t, r 2 = R 2 p β p βt t γ s γs, r 3 = R 3 δ s δs where p i s ad i s are distict sets of pries. By perutig r ad r 3 if ecessary, we ay assue that all the p i are odd. We select a uber a 0 coprie to as follows. Suppose R R 2 (od p β p βt t γ s γs 3 ).
14 The both the ubers R ± p β + are ot cogruet to R 2 (od the is coprie to. We take a 0 to be oe of R ± (od p β pβ t t γ γs s ivolvig f(r 2 ). Thus we get p β + p β pβ t t γ γs s ) ad oe of which is coprie to. If R R 2 ), the we take a 0 = R. Hece i (5) with a = a 0, o ter occurs 0 = Af(r ) + Bf(r 3 ) where A ad B represet certai o-vaishig series of positive ters. Thus f(r ) < 0 which f(r 3 ) is a cotradictio with f(r ) < 0, f(r 2) < 0. f(r 2 ) f(r 3 ) Proof of Theore 2. We assue without loss of geerality that gcd (, s, s 2, s 3 ) =. By usig partial fractios, we have where A = We observe that U = A { + + s B + s 2 + C + s 3 } β αs (s s 2 )(s s 3 ), B = β αs 2 (s 2 s )(s 2 s 3 ), C = β αs 3 (s 3 s )(s 3 s 2 ). (24) A + B + C = 0. Let s i r i (od ) with 0 < r i for i =, 2, 3. The we re-write (25) U = γ + A { + + r B + r 2 + C + r 3 } where γ Q(α, β). Suppose r = r 2. The r r 3 ad by (24), we have (26) U = γ + C(r r 3 ) ( + r )( + r 3 ). By our assuptio C 0. Hece the ifiite su i (26) does ot vaish. Now we apply [, Theore ] to coclude that U is trascedetal. Siilarly U is trascedetal wheever r = r 3 or r 2 = r 3. Thus we ay assue that r, r 2, r 3 are all distict. Now we apply Lea 3 to coclude that the ifiite su i (25) ad hece U is trasscedetal. We observe fro (22),(23),(24) that i the cases whe s, s 2, s 3 are all i the sae residue class (od ) or whe (22) is ot valid, the U is algebraic. 4
15 Refereces. [] S.D. Adhikari, N. Saradha, T.N. Shorey, R. Tijdea, Trascedetal ifiite sus, Idag. Math. N.S. 2 (200), -4. [2] S.D. Adhikari, Trascedetal Ifiite sus ad related uestios, to appear i Proc. Iter. Cof. o Nuber Theory at Chadigarh, October [3] A. Baker, B.J. Birch, E.A. Wirsig, O a proble of Chowla, J. Nuber Th. 5 (973), [4] S. Chowla, The Riea zeta ad allied fuctios, Bull. Aer. Math. Soc. 58(952), [5] A.E. Livigsto, The series f()/ for periodic f, Caad. Math. Bull. 8 (965), [6] T. Okada, O a certai ifiite sus for a periodic arithetical fuctios, Acta Arith. 40 (982), [7] R. Tijdea, Soe applicatios of diophatie approxiatios, to appear i Proc. Milleiu Cof. o Nuber Theory at Urbaa, May N. Saradha R. Tijdea School of Matheatics Matheatical Istitute Tata Istitute of Fudaetal Research Leide Uiversity Hoi Bhabha Road P.O. Box 952 Mubai RA Leide Idia The Netherlads eail: saradha@ath.tifr.res.i eail: tijdea@ath.leideuiv.l 5
Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a
Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi
More informationFACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS. Dedicated to the memory of Paul Erdős. 1. Introduction. n k. f n,a =
FACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS NEIL J. CALKIN Abstract. We prove divisibility properties for sus of powers of bioial coefficiets ad of -bioial coefficiets. Dedicated to the eory of
More informationBernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes
Beroulli Polyoials Tals give at LSBU, October ad Noveber 5 Toy Forbes Beroulli Polyoials The Beroulli polyoials B (x) are defied by B (x), Thus B (x) B (x) ad B (x) x, B (x) x x + 6, B (x) dx,. () B 3
More information18.S34 (FALL, 2007) GREATEST INTEGER PROBLEMS. n + n + 1 = 4n + 2.
18.S34 (FALL, 007) GREATEST INTEGER PROBLEMS Note: We use the otatio x for the greatest iteger x, eve if the origial source used the older otatio [x]. 1. (48P) If is a positive iteger, prove that + + 1
More informationBertrand s postulate Chapter 2
Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are
More information42 Dependence and Bases
42 Depedece ad Bases The spa s(a) of a subset A i vector space V is a subspace of V. This spa ay be the whole vector space V (we say the A spas V). I this paragraph we study subsets A of V which spa V
More informationA PROBABILITY PROBLEM
A PROBABILITY PROBLEM A big superarket chai has the followig policy: For every Euros you sped per buy, you ear oe poit (suppose, e.g., that = 3; i this case, if you sped 8.45 Euros, you get two poits,
More informationAutomated Proofs for Some Stirling Number Identities
Autoated Proofs for Soe Stirlig Nuber Idetities Mauel Kauers ad Carste Scheider Research Istitute for Sybolic Coputatio Johaes Kepler Uiversity Altebergerstraße 69 A4040 Liz, Austria Subitted: Sep 1, 2007;
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationLOWER BOUNDS FOR MOMENTS OF ζ (ρ) 1. Introduction
LOWER BOUNDS FOR MOMENTS OF ζ ρ MICAH B. MILINOVICH AND NATHAN NG Abstract. Assuig the Riea Hypothesis, we establish lower bouds for oets of the derivative of the Riea zeta-fuctio averaged over the otrivial
More informationSummer MA Lesson 13 Section 1.6, Section 1.7 (part 1)
Suer MA 1500 Lesso 1 Sectio 1.6, Sectio 1.7 (part 1) I Solvig Polyoial Equatios Liear equatio ad quadratic equatios of 1 variable are specific types of polyoial equatios. Soe polyoial equatios of a higher
More informationBinomial transform of products
Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {
More informationGENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES
J. Nuber Theory 0, o., 9-9. GENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES Zhi-Hog Su School of Matheatical Scieces, Huaiyi Noral Uiversity, Huaia, Jiagsu 00, PR Chia Eail: zhihogsu@yahoo.co
More informationSOME FINITE SIMPLE GROUPS OF LIE TYPE C n ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER
Joural of Algebra, Nuber Theory: Advaces ad Applicatios Volue, Nuber, 010, Pages 57-69 SOME FINITE SIMPLE GROUPS OF LIE TYPE C ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER School
More informationLecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces
Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such
More informationMetric Dimension of Some Graphs under Join Operation
Global Joural of Pure ad Applied Matheatics ISSN 0973-768 Volue 3, Nuber 7 (07), pp 333-3348 Research Idia Publicatios http://wwwripublicatioco Metric Diesio of Soe Graphs uder Joi Operatio B S Rawat ad
More informationSome remarks on the paper Some elementary inequalities of G. Bennett
Soe rears o the paper Soe eleetary iequalities of G. Beett Dag Ah Tua ad Luu Quag Bay Vieta Natioal Uiversity - Haoi Uiversity of Sciece Abstract We give soe couterexaples ad soe rears of soe of the corollaries
More information[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.
[ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural
More informationMath 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version]
Math 4707 Sprig 08 Darij Griberg: idter page Math 4707 Sprig 08 Darij Griberg: idter with solutios [preliiary versio] Cotets 0.. Coutig first-eve tuples......................... 3 0.. Coutig legal paths
More informationOn Order of a Function of Several Complex Variables Analytic in the Unit Polydisc
ISSN 746-7659, Eglad, UK Joural of Iforatio ad Coutig Sciece Vol 6, No 3, 0, 95-06 O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc Rata Kuar Dutta + Deartet of Matheatics, Siliguri
More informationarxiv: v1 [math.nt] 26 Feb 2014
FROBENIUS NUMBERS OF PYTHAGOREAN TRIPLES BYUNG KEON GIL, JI-WOO HAN, TAE HYUN KIM, RYUN HAN KOO, BON WOO LEE, JAEHOON LEE, KYEONG SIK NAM, HYEON WOO PARK, AND POO-SUNG PARK arxiv:1402.6440v1 [ath.nt] 26
More informationx !1! + 1!2!
4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio
More informationBINOMIAL COEFFICIENT HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES
#A37 INTEGERS (20) BINOMIAL COEFFICIENT HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES Derot McCarthy Departet of Matheatics, Texas A&M Uiversity, Texas ccarthy@athtauedu Received: /3/, Accepted:
More informationTHE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION
MATHEMATICA MONTISNIGRI Vol XXVIII (013) 17-5 THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION GLEB V. FEDOROV * * Mechaics ad Matheatics Faculty Moscow State Uiversity Moscow, Russia
More informationX. Perturbation Theory
X. Perturbatio Theory I perturbatio theory, oe deals with a ailtoia that is coposed Ĥ that is typically exactly solvable of two pieces: a referece part ad a perturbatio ( Ĥ ) that is assued to be sall.
More informationA New Type of q-szász-mirakjan Operators
Filoat 3:8 07, 567 568 https://doi.org/0.98/fil7867c Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat A New Type of -Szász-Miraka Operators
More informationTHE ARTIN CARMICHAEL PRIMITIVE ROOT PROBLEM ON AVERAGE
THE ARTIN CARMICHAEL PRIMITIVE ROOT PROBLEM ON AVERAGE SHUGUANG LI AND CARL POMERANCE Abstract. For a atural uber, let λ() deote the order of the largest cyclic subgroup of (Z/Z). For a give iteger a,
More informationCERTAIN CONGRUENCES FOR HARMONIC NUMBERS Kotor, Montenegro
MATHEMATICA MONTISNIGRI Vol XXXVIII (017) MATHEMATICS CERTAIN CONGRUENCES FOR HARMONIC NUMBERS ROMEO METROVIĆ 1 AND MIOMIR ANDJIĆ 1 Maritie Faculty Kotor, Uiversity of Moteegro 85330 Kotor, Moteegro e-ail:
More informationIRRATIONALITY MEASURES, IRRATIONALITY BASES, AND A THEOREM OF JARNÍK 1. INTRODUCTION
IRRATIONALITY MEASURES IRRATIONALITY BASES AND A THEOREM OF JARNÍK JONATHAN SONDOW ABSTRACT. We recall that the irratioality expoet µα ( ) of a irratioal umber α is defied usig the irratioality measure
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationA new sequence convergent to Euler Mascheroni constant
You ad Che Joural of Iequalities ad Applicatios 08) 08:7 https://doi.org/0.86/s3660-08-670-6 R E S E A R C H Ope Access A ew sequece coverget to Euler Mascheroi costat Xu You * ad Di-Rog Che * Correspodece:
More informationRiemann Hypothesis Proof
Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. Riea Hypothesis Proof H. Vic Dao vic0@cocast.et March, 009 Revised Deceber, 009 Abstract
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationM17 MAT25-21 HOMEWORK 5 SOLUTIONS
M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series
More informationTeacher s Marking. Guide/Answers
WOLLONGONG COLLEGE AUSRALIA eacher s Markig A College of the Uiversity of Wollogog Guide/Aswers Diploa i Iforatio echology Fial Exaiatio Autu 008 WUC Discrete Matheatics his exa represets 60% of the total
More informationk-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction
Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)
More informationOrthogonal Functions
Royal Holloway Uiversity of odo Departet of Physics Orthogoal Fuctios Motivatio Aalogy with vectors You are probably failiar with the cocept of orthogoality fro vectors; two vectors are orthogoal whe they
More informationdistinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k)
THE TWELVEFOLD WAY FOLLOWING GIAN-CARLO ROTA How ay ways ca we distribute objects to recipiets? Equivaletly, we wat to euerate equivalece classes of fuctios f : X Y where X = ad Y = The fuctios are subject
More informationProblem. Consider the sequence a j for j N defined by the recurrence a j+1 = 2a j + j for j > 0
GENERATING FUNCTIONS Give a ifiite sequece a 0,a,a,, its ordiary geeratig fuctio is A : a Geeratig fuctios are ofte useful for fidig a closed forula for the eleets of a sequece, fidig a recurrece forula,
More informationName Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions
Nae Period ALGEBRA II Chapter B ad A Notes Solvig Iequalities ad Absolute Value / Nubers ad Fuctios SECTION.6 Itroductio to Solvig Equatios Objectives: Write ad solve a liear equatio i oe variable. Solve
More informationCommon Fixed Points for Multifunctions Satisfying a Polynomial Inequality
BULETINUL Uiversităţii Petrol Gaze di Ploieşti Vol LXII No /00 60-65 Seria Mateatică - Iforatică - Fizică Coo Fixed Poits for Multifuctios Satisfyig a Polyoial Iequality Alexadru Petcu Uiversitatea Petrol-Gaze
More informationInternational Journal of Mathematical Archive-4(9), 2013, 1-5 Available online through ISSN
Iteratioal Joural o Matheatical Archive-4(9), 03, -5 Available olie through www.ija.io ISSN 9 5046 THE CUBIC RATE OF CONVERGENCE OF GENERALIZED EXTRAPOLATED NEWTON RAPHSON METHOD FOR SOLVING NONLINEAR
More informationDIRICHLET CHARACTERS AND PRIMES IN ARITHMETIC PROGRESSIONS
DIRICHLET CHARACTERS AND PRIMES IN ARITHMETIC PROGRESSIONS We la to rove the followig Theore (Dirichlet s Theore) Let (a, k) = The the arithetic rogressio cotais ifiitely ay ries a + k : = 0,, 2, } = :
More informationA talk given at Institut Camille Jordan, Université Claude Bernard Lyon-I. (Jan. 13, 2005), and University of Wisconsin at Madison (April 4, 2006).
A tal give at Istitut Caille Jorda, Uiversité Claude Berard Lyo-I (Ja. 13, 005, ad Uiversity of Wiscosi at Madiso (April 4, 006. SOME CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS Zhi-Wei Su Departet
More informationf(1), and so, if f is continuous, f(x) = f(1)x.
2.2.35: Let f be a additive fuctio. i Clearly fx = fx ad therefore f x = fx for all Z+ ad x R. Hece, for ay, Z +, f = f, ad so, if f is cotiuous, fx = fx. ii Suppose that f is bouded o soe o-epty ope set.
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationTopics. Homework Problems. MATH 301 Introduction to Analysis Chapter Four Sequences. 1. Definition of convergence of sequences.
MATH 301 Itroductio to Aalysis Chapter Four Sequeces Topics 1. Defiitio of covergece of sequeces. 2. Fidig ad provig the limit of sequeces. 3. Bouded covergece theorem: Theorem 4.1.8. 4. Theorems 4.1.13
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationRefinements of Jensen s Inequality for Convex Functions on the Co-Ordinates in a Rectangle from the Plane
Filoat 30:3 (206, 803 84 DOI 0.2298/FIL603803A Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat Refieets of Jese s Iequality for Covex
More informationDIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS
DIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS VERNER E. HOGGATT, JR. Sa Jose State Uiversity, Sa Jose, Califoria 95192 ad CALVIN T. LONG Washigto State Uiversity, Pullma, Washigto 99163
More information[ 47 ] then T ( m ) is true for all n a. 2. The greatest integer function : [ ] is defined by selling [ x]
[ 47 ] Number System 1. Itroductio Pricile : Let { T ( ) : N} be a set of statemets, oe for each atural umber. If (i), T ( a ) is true for some a N ad (ii) T ( k ) is true imlies T ( k 1) is true for all
More information6. Uniform distribution mod 1
6. Uiform distributio mod 1 6.1 Uiform distributio ad Weyl s criterio Let x be a seuece of real umbers. We may decompose x as the sum of its iteger part [x ] = sup{m Z m x } (i.e. the largest iteger which
More informationJORGE LUIS AROCHA AND BERNARDO LLANO. Average atchig polyoial Cosider a siple graph G =(V E): Let M E a atchig of the graph G: If M is a atchig, the a
MEAN VALUE FOR THE MATCHING AND DOMINATING POLYNOMIAL JORGE LUIS AROCHA AND BERNARDO LLANO Abstract. The ea value of the atchig polyoial is coputed i the faily of all labeled graphs with vertices. We dee
More informationDomination Number of Square of Cartesian Products of Cycles
Ope Joural of Discrete Matheatics, 01,, 88-94 Published Olie October 01 i SciRes http://wwwscirporg/joural/ojd http://dxdoiorg/10436/ojd014008 Doiatio Nuber of Square of artesia Products of ycles Morteza
More informationDirichlet s Theorem on Arithmetic Progressions
Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty
More information1.2 AXIOMATIC APPROACH TO PROBABILITY AND PROPERTIES OF PROBABILITY MEASURE 1.2 AXIOMATIC APPROACH TO PROBABILITY AND
NTEL- robability ad Distributios MODULE 1 ROBABILITY LECTURE 2 Topics 1.2 AXIOMATIC AROACH TO ROBABILITY AND ROERTIES OF ROBABILITY MEASURE 1.2.1 Iclusio-Exclusio Forula I the followig sectio we will discuss
More informationMATH 304: MIDTERM EXAM SOLUTIONS
MATH 304: MIDTERM EXAM SOLUTIONS [The problems are each worth five poits, except for problem 8, which is worth 8 poits. Thus there are 43 possible poits.] 1. Use the Euclidea algorithm to fid the greatest
More informationChapter 2. Asymptotic Notation
Asyptotic Notatio 3 Chapter Asyptotic Notatio Goal : To siplify the aalysis of ruig tie by gettig rid of details which ay be affected by specific ipleetatio ad hardware. [1] The Big Oh (O-Notatio) : It
More informationA string of not-so-obvious statements about correlation in the data. (This refers to the mechanical calculation of correlation in the data.
STAT-UB.003 NOTES for Wedesday 0.MAY.0 We will use the file JulieApartet.tw. We ll give the regressio of Price o SqFt, show residual versus fitted plot, save residuals ad fitted. Give plot of (Resid, Price,
More informationSome results on the Apostol-Bernoulli and Apostol-Euler polynomials
Soe results o the Apostol-Beroulli ad Apostol-Euler polyoials Weipig Wag a, Cagzhi Jia a Tiaig Wag a, b a Departet of Applied Matheatics, Dalia Uiversity of Techology Dalia 116024, P. R. Chia b Departet
More informationTransfer Function Analysis
Trasfer Fuctio Aalysis Free & Forced Resposes Trasfer Fuctio Syste Stability ME375 Trasfer Fuctios - Free & Forced Resposes Ex: Let s s look at a stable first order syste: τ y + y = Ku Take LT of the I/O
More informationExpansion of the integral x. integral from the value x = 0 to x = 1 * f 1 dx(log x) m n having extended the. Leonhard Euler THEOREM 1 PROOF.
Epasio of the itegral f dlog havig eteded the itegral fro the value = 0 to = * Leohard Euler THEOREM If deotes a positive iteger ad the itegral f d g is eteded fro the value = 0 to = the value of the itegral
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationn=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n
Series. Defiitios ad first properties A series is a ifiite sum a + a + a +..., deoted i short by a. The sequece of partial sums of the series a is the sequece s ) defied by s = a k = a +... + a,. k= Defiitio
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationSchool of Mechanical Engineering Purdue University. ME375 Transfer Functions - 1
Trasfer Fuctio Aalysis Free & Forced Resposes Trasfer Fuctio Syste Stability ME375 Trasfer Fuctios - 1 Free & Forced Resposes Ex: Let s look at a stable first order syste: y y Ku Take LT of the I/O odel
More informationComplete Solutions to Supplementary Exercises on Infinite Series
Coplete Solutios to Suppleetary Eercises o Ifiite Series. (a) We eed to fid the su ito partial fractios gives By the cover up rule we have Therefore Let S S A / ad A B B. Covertig the suad / the by usig
More informationOn The Prime Numbers In Intervals
O The Prie Nubers I Itervals arxiv:1706.01009v1 [ath.nt] 4 Ju 2017 Kyle D. Balliet A Thesis Preseted to the Faculty of the Departet of Matheatics West Chester Uiversity West Chester, Pesylvaia I Partial
More informationMath 140A Elementary Analysis Homework Questions 1
Math 14A Elemetary Aalysis Homewor Questios 1 1 Itroductio 1.1 The Set N of Natural Numbers 1 Prove that 1 2 2 2 2 1 ( 1(2 1 for all atural umbers. 2 Prove that 3 11 (8 5 4 2 for all N. 4 (a Guess a formula
More information19.1 The dictionary problem
CS125 Lecture 19 Fall 2016 19.1 The dictioary proble Cosider the followig data structural proble, usually called the dictioary proble. We have a set of ites. Each ite is a (key, value pair. Keys are i
More information(s)h(s) = K( s + 8 ) = 5 and one finite zero is located at z 1
ROOT LOCUS TECHNIQUE 93 should be desiged differetly to eet differet specificatios depedig o its area of applicatio. We have observed i Sectio 6.4 of Chapter 6, how the variatio of a sigle paraeter like
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationPerturbation Theory, Zeeman Effect, Stark Effect
Chapter 8 Perturbatio Theory, Zeea Effect, Stark Effect Ufortuately, apart fro a few siple exaples, the Schrödiger equatio is geerally ot exactly solvable ad we therefore have to rely upo approxiative
More informationAvailable online at J. Math. Comput. Sci. 4 (2014), No. 3, ISSN:
Available olie at http://scik.org J. Math. Coput. Sci. (1, No. 3, 9-5 ISSN: 197-537 ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION FUAD. S. M. AL SARARI 1,, S. LATHA 1 Departet of Studies i Matheatics,
More informationq-fibonacci polynomials and q-catalan numbers Johann Cigler [ ] (4) I don t know who has observed this well-known fact for the first time.
-Fiboacci polyoials ad -Catala ubers Joha Cigler The Fiboacci polyoials satisfy the recurrece F ( x s) = s x = () F ( x s) = xf ( x s) + sf ( x s) () with iitial values F ( x s ) = ad F( x s ) = These
More informationStochastic Matrices in a Finite Field
Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices
More informationSZEGO S THEOREM STARTING FROM JENSEN S THEOREM
UPB Sci Bull, Series A, Vol 7, No 3, 8 ISSN 3-77 SZEGO S THEOREM STARTING FROM JENSEN S THEOREM Cǎli Alexe MUREŞAN Mai îtâi vo itroduce Teorea lui Jese şi uele coseciţe ale sale petru deteriarea uǎrului
More informationA Note on the Symmetric Powers of the Standard Representation of S n
A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationOn twin primes associated with the Hawkins random sieve
Joural of Nuber Theory 9 006 84 96 wwwelsevierco/locate/jt O twi pries associated with the Hawkis rado sieve HM Bui,JPKeatig School of Matheatics, Uiversity of Bristol, Bristol, BS8 TW, UK Received 3 July
More informationSection 5.5. Infinite Series: The Ratio Test
Differece Equatios to Differetial Equatios Sectio 5.5 Ifiite Series: The Ratio Test I the last sectio we saw that we could demostrate the covergece of a series a, where a 0 for all, by showig that a approaches
More informationBasic Sets. Functions. MTH299 - Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S.
Basic Sets Example 1. Let S = {1, {2, 3}, 4}. Idicate whether each statemet is true or false. (a) S = 4 (b) {1} S (c) {2, 3} S (d) {1, 4} S (e) 2 S. (f) S = {1, 4, {2, 3}} (g) S Example 2. Compute the
More informationBOUNDS ON SOME VAN DER WAERDEN NUMBERS. Tom Brown Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6
BOUNDS ON SOME VAN DER WAERDEN NUMBERS To Brow Departet of Matheatics, Sio Fraser Uiversity, Buraby, BC V5A S6 Bruce M Lada Departet of Matheatics, Uiversity of West Georgia, Carrollto, GA 308 Aaro Robertso
More informationDouble Derangement Permutations
Ope Joural of iscrete Matheatics, 206, 6, 99-04 Published Olie April 206 i SciRes http://wwwscirporg/joural/ojd http://dxdoiorg/04236/ojd2066200 ouble erageet Perutatios Pooya aeshad, Kayar Mirzavaziri
More informationON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS
ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS NORBERT KAIBLINGER Abstract. Results of Lid o Lehmer s problem iclude the value of the Lehmer costat of the fiite cyclic group Z/Z, for 5 ad all odd. By complemetary
More informationSOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR
Joural of the Alied Matheatics Statistics ad Iforatics (JAMSI) 5 (9) No SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR SP GOYAL AND RAKESH KUMAR Abstract Here we
More informationINMO-2018 problems and solutions
Pioeer Educatio The Best Way To Success INMO-018 proles ad solutios NTSE Olypiad AIPMT JEE - Mais & Advaced 1. Let ABC e a o-equilateral triagle with iteger sides. Let D ad E e respectively the id-poits
More informationCOMP 2804 Solutions Assignment 1
COMP 2804 Solutios Assiget 1 Questio 1: O the first page of your assiget, write your ae ad studet uber Solutio: Nae: Jaes Bod Studet uber: 007 Questio 2: I Tic-Tac-Toe, we are give a 3 3 grid, cosistig
More informationAl Lehnen Madison Area Technical College 10/5/2014
The Correlatio of Two Rado Variables Page Preliiary: The Cauchy-Schwarz-Buyakovsky Iequality For ay two sequeces of real ubers { a } ad = { b } =, the followig iequality is always true. Furtherore, equality
More informationREVIEW OF CALCULUS Herman J. Bierens Pennsylvania State University (January 28, 2004) x 2., or x 1. x j. ' ' n i'1 x i well.,y 2
REVIEW OF CALCULUS Hera J. Bieres Pesylvaia State Uiversity (Jauary 28, 2004) 1. Suatio Let x 1,x 2,...,x e a sequece of uers. The su of these uers is usually deoted y x 1 % x 2 %...% x ' j x j, or x 1
More informationGeneralized Fibonacci-Like Sequence and. Fibonacci Sequence
It. J. Cotep. Math. Scieces, Vol., 04, o., - 4 HIKARI Ltd, www.-hiari.co http://dx.doi.org/0.88/ijcs.04.48 Geeralized Fiboacci-Lie Sequece ad Fiboacci Sequece Sajay Hare epartet of Matheatics Govt. Holar
More informationPROBLEM SET 5 SOLUTIONS 126 = , 37 = , 15 = , 7 = 7 1.
Math 7 Sprig 06 PROBLEM SET 5 SOLUTIONS Notatios. Give a real umber x, we will defie sequeces (a k ), (x k ), (p k ), (q k ) as i lecture.. (a) (5 pts) Fid the simple cotiued fractio represetatios of 6
More informationRoger Apéry's proof that zeta(3) is irrational
Cliff Bott cliffbott@hotmail.com 11 October 2011 Roger Apéry's proof that zeta(3) is irratioal Roger Apéry developed a method for searchig for cotiued fractio represetatios of umbers that have a form such
More informationThe Riemann Zeta Function
Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we
More informationA GENERALIZED BERNSTEIN APPROXIMATION THEOREM
Ø Ñ Å Ø Ñ Ø Ð ÈÙ Ð Ø ÓÒ DOI: 10.2478/v10127-011-0029-x Tatra Mt. Math. Publ. 49 2011, 99 109 A GENERALIZED BERNSTEIN APPROXIMATION THEOREM Miloslav Duchoň ABSTRACT. The preset paper is cocered with soe
More informationTranscendental values of the digamma function
Joural of Number Theory 25 (2007) 298 38 www.elsevier.com/locate/jt Trascedetal values of the digamma fuctio M. Ram Murty a,,,n.saradha b a Departmet of Mathematics, Quee s Uiversity, Kigsto, Otario, K7L
More informationA Pair of Operator Summation Formulas and Their Applications
A Pair of Operator Suatio Forulas ad Their Applicatios Tia-Xiao He 1, Leetsch C. Hsu, ad Dogsheg Yi 3 1 Departet of Matheatics ad Coputer Sciece Illiois Wesleya Uiversity Blooigto, IL 6170-900, USA Departet
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More information