CONVERGENCE OF THE COEFRCIENTS IN A RECURRING POWER SERIES JOSEPH ARK IN Nanuet, New York 1. INTRODUCTION. In this paper we use the following notation

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1 CONVERGENCE OF THE COEFRCIENTS IN A RECURRING POWER SERIES JOSEPH ARK IN Nauet, Ne York 1. INTRODUCTION I this paper e use the folloig otatio oo \ k oo 1 \ ^ (k) 1 _ X c x Ev* -E k = / = (For coveiece, e shall rite c istead of c.) We defie Ev = F ( x ) ^ = for a fiite f, t m d v = TT< i -v> = «<*> W= W=l for fiite t ad m, here the d ^ ad are positive itegers. The r fi ad are distict ad e say Jr^ is the greatest r i the jr J. If 2. THEOREM 1 F(x)/Q(x) = 2 ] u x, = 41

2 42 CONVERGENCE OF THE COEFFICIENTS [Feb. the (2.1) lim iu / u.1 (for a fiite j =,1,2,' ) J *>oo j / -j j coverges to rj 9 here the r ^ i Q(x) are distict ith distict moduli ad r A j is the greatest j r i the J r I. Proof* It has bee sho by Poicare [ 1 ] that (2.2) lim u / u, -^>oo / -i coverges to some root (r) i Q(x). i Q(x) is Let (rjj.) (We must the prove that this root (r) m (2.3) MW = J J (l-r x) W, =i here the p are positive itegers or = ad di + P i = d 2 + p 2 =... - p + d = k (k = 1, 2 9 3, ) for a fiite = 1, 2, 3,, m. The* m M(x)Q(x) = " P J (1 - r x ) k =i = k (x) so that (2.4) F(x)M(x)/Q(x)M(x) = F(x)M(x)/<^(x) - Ev* -!> *' =o =o

3 1969] IN A RECURRING POWER SERIES 43 here it is evidet u = c(k,). No let 4 ( x ) = \ = c (here v is fiite), here combiig this ith (2,4), e rite (2.5) F W M W / ^ W - ^ c(k.- 1, )x W oo. = V= / \= ad combiig coefficiets leads to v (2.5.1) cfc-l.) =^cfc-)c = V v W= W= c. k = 2, 3, 4, -. I (2.5.1), e replace k ith k + 1 (here k = 1,2,3, ) here combiig this result ith (2.2) leads to 11m S l ^ o o i c(k + l,)/c(k + l, - 1)1 coverges to some root (r) i Q(x). B For coveiece, e rite the covergece as (2.5.2) c ( k + l, ) = g k + i c ( k + l, - l ).

4 44 CONVERGENCE OF THE COEFFICIENTS [Feb. Combiig (2.5.1) ith k replaced by k + 1 ith (2.5.2), it is easily sho, that for a fiite v, e have (2.5.3) c(k,)/c(k,- 1) = g k v / v = y c ( k + l, - ) c iy c(k + l, - - l ) c = / = = V i > so that (2-5.4) g k + 1 = g k =... = g I. Thus to complete the proof of Theorem 1, it remais to sho that The e cosider the folloig (e refer to (2.3) ) m oo (2.6) (^(x))" 1 = T T (1 - i ^ x ) " 1 = Y ^ e(m,)x (for a fiite m) W=l W= for the covergece properties of e(m, )/e(m, - 1) 9 here the lr distict ad r j is the greatest root NOTE. For coveiece, e rite I W j j are e(m, )/e(m, - j) = ir (for a fiite j =,1, 2, ), i place of I e(m,)/e(m, - j) j coverges to \r j,

5 19^9] IN A RECURRING POWER SERIES For m = 1, e have (2.7) ( i» x)" 1 = ^ e(1 > > x W > = here e(l, ) = r, so that e ( l, ) / e ( l, - j ) = rj For m = 2, e have (2.8) [(1 - r lx ) (1 - rax)]"" 1 = ^ e(2,) = X here e(2 9 ) = (r ' - r )/(v t - r 2 ), so that e(2,)/e(2, - j) = rj It o remais to cosider for fiite m = 3, 4, 5, e / ^ V 1 t s= s=l for a fiite t = 8,4, 5, * 8, here UQ = 1.

6 46 CONVERGENCE OF THE COEFFICIENTS [Feb. Equatig the coefficiets i this leads to (2.1) U = X X ~ s V s s=l t (U = 1 > ad t-1 Ui = U a tral, U 2 = Uia t r a l + U a t ^ % U t = ^ U ^. Also, sice i (2.9), e have e may rite s= t t-1 TTVv> = i -E a s xt " s s=l s= t t-1 (2.1D T f V v = x t - Z v s s=l s= We o combie (2.1) ith (2.11) ad rite (2.12) - ^ + s-1 E ( v E V H r - t-s 8 ) x s=2 V r=l r Multiplyig (2*12) by x ad combiig the result ith t-1 Uixt = Ui S s s=

7 1969] IN A RECURRING POWER SERIES i (2.11) leads to 47 (2.13) x t+i U 2 x t " 1 + E ( U r E U r ^ s V s ^ j r=o s=o ' t-r-2 + Utao. No, multiplyig (2.13) by x ad combiig the r e s u l t ith ^ s=o a s x i (2.11) 9 e the have (2.14) x t+2 U ^ 1 + E ( U r h l - Z U r t - s V s - i ). tr-y-2 r=o + a U 2 We cotiue i the exact ay e foud (2.13) ad (2.14) for - 1 steps to get t-3 (2.15) J**- 1 - U x M + 5 ] r=o u. x t - r - 2 +r+1 JLJ + r - s a t - s - i I s=o ' > i + U au = U x ' R(x) + U a.. -i -i o We o cotiue (2.15) ith (2.11) to get the folloig t equatios r ' +_i = V*" 1 + R(ri) + Vl a ' (2.16) rt+-i = t-l + R ( } t

8 48 CONVERGENCE OF THE COEFFICIENTS [Feb. Next, e cosider the t equatios obtaied from (2.16). These t equatios i the t uko ca be solved by C r a m e r f s rule to obtai (2.17) U D 2 = D l ( ), here Djdi) ad D 2 a r e the determiats give belo: (2.18) Dt() = - t+- t+- r t -1-1 t-2 *r : - ' r i r t 1 1 (2.19) D, = t~i r j r* t-2 r i t-2 r t " r j r t l l 1 We o replace ith - 1 i (2.17) to get (2.2) IT DL 2 = Dj 1 ( - 1), - i ad dividig (2.17) by (2.2), e get (2.21) U / U - i = D l ( ) / D l ( " 1] Sice the r tha the other roots, ad e rite ^ ad a r e distict, the oe root (say r i is g r e a t e r (2.22) U / U - i = t+-2 ^ t+-2^ ( D i ( ) / r p ^ ) / ( D 1 ( - l ) / r r 1 1 ^ ) t+ 7 No i (2.22) e let r i ' (i the umerator) divide every t e r m of the first t+ 2 colum i (2.18) ad r A (i the deomiator) divide every t e r m i the first colum of (2.18) (ith replaced by - 1). The if e let ->oo it i s e v i - det that

9 1969] (2.23) IN A RECURRING POWER SERIES lim ->oo tu /U - i! = I r 1 J. 49 No for a fiite t e rite lim -> oo I U -j. /U -j-i. r J (j =,1, 2,, t - 1), so that (2.24) lim ^ool U /U -tl J Multiplyig the F(x) i (1) ith uu S= i (2.9), e rite (2.25) =o / \ s=o / s=o C x S, s here comparig the coefficiets e have (2.26) Z-^ -i s s=o b s No, sice f is fiite, ad by the results i (2.23), e rite C = ri Y ^ U b = ri C 1 * / J -s-i s -i * s=o

10 5 CONVERGENCE OF THE COEFFICIENTS [Feb. here combiig this ith the r ^ ad are distict (so that e may add that the r, have distict moduli), leads to the completio of the proof for Theorem 1. From (2.7), (2.8), ad (2.17), the folloig corollary is immediate: Corollary. If t s=i s=o here the r s ^ ad are distict, the (2.27) It is alays possible to solve for the U ( =,1, 2,* ) Let as a fuctio of the r. s -k SECTION 3 t =i / =l =o oo (k) c x (CQ r = 1 ad k = 1, 2, 3, ) for a fiite t = 2, 3,4, ad the give roots ^ ad are distict. We also defie s(x) t t = Z E a ^r^+-r" - c ~ xw_1 = =i r= ad b = Z a x 7 W-2 a ;" W=2

11 19691 IN A RECURRING POWER SERIES 51 here *Xi ^ ad is a root i S(x) = 8(x t ) =. We the have the folloig: Theorem 2. If c = 1, ci = aic, c 2 = atcj + a 2 c, e e t-i 3 t ~ J L +i t--i =o ad p.. = a 4 (k + - j) (j = 1, 2, 3,, ), the q m+i = b * " m ^ 2 k + ~ m ~ ^ (m = l, 2, 3, - - -, - l ) (3.1) c ( k ) / c ( k ) = E /G (k, = 1, 2, 3, ), 9 ' -i ' here E ad G are the determiats give belo. Pi 2-1 P2 qa (3.1.1) E = - 1 P3 - i q 4 P4 q 5 - i p - l % - 1 p *It should be oted that sice the a! s are costat for a fixed t, that the root xi ill be determied as a variable* sice it is a fuctio of the c ad ill, of course, chage values for differet -

12 52 CONVERGENCE OF THE COEFFICIENTS [Feb P2 qs o o o 1-1 P3 Q4 (3.1.2) G = -1 p 4 q 5-1 p 5 q 6 Proof. Let (3.2) 1 = 1 IvjfE W=l ' \ W= C X - 1 V i \ \ -1 p (for a fiite ), here the a ad the c are idetical to those i (3). The multiplyig ad combiig the terms i (3.2) leads to S(xi) = S(x) = i (3). No, takig each side of (3.2) to the k poer, e rite (3.3) 1" = 1 E =i a x vk v=o ( k = 2, 3, - - ), (here, of course, xj is a root i (3.3) ). Usig the correspodig values i (3), e rite (3.3) as (3.3.1) 1 = (1 - aix - bx' \=o ( k ) X W + J ( K ) Differetiatio of (3.3.1) leads to k(a lx + 2bx 2 ) I ^ c^k) + J(x) j = (1 - a x x - bx* )l ^ c k) x + W(x) <=o \ = i

13 1969] IN A RECURRING POWER SERIES 53 ad by comparig coefficiets, e coclude that (3.4) c^k) = a i ( k + - l ) c ^ + b(2k + - 2)cf^ for k = 2, 3,..., = 2, 3, -, e (k) = 1 ad c k ) = ajk. (k) Whe e divide (3.4) by c, e get c ( k ) /i. i\ _. M2k + - 2)( - 1) = a! ( k + - l ) + Tgr,. v (,k = 2, 3, ), c ( - l)c - i ^ ~ -2 - i hich i tur, alog ith CQ = 1 ad ci = aik, implies (alog ith the values of p ad q i (3)) 9 (k) (3 c q 2 qs,,, q - i. %. - 5) Jr =^ + ^ +?r + - +^:: + ^ = K() - V i - - i ' We complete the proof of T h e o r e m 2 ith E u l e r f s statemet [2] K W = E / G ; ad e resolve for the case he k = 1 ith (2.27). Corollary. I it is alays possible to solve for TT«-v»- k -h-zv = 1+ 1 \ t \ - k -, ^ (k) = 1 + > e x =i \ =l / =i

14 54 CONVERGENCE OF THE COEFFICIENTS [Feb. (3.6) c k ) / c S i = K ( ) = E / G ( k a d = 2 s 3 f ' ' * ) he t = 2, 3,4, or 5, if the r ^ ad are distict. Proof. I (2.27), it is see that the c maybe determied. No, sice t - 1 = 1, 2, 3, or 4, the the roots (each root is a fuctio of the c ) i S(x).(i 3) may alays be foud, so that e ill obtai values for the p q. We the complete the proof of the corollary by observig that E ad G are both fuctios of the p ad q. I coclusio: We solve he t = 1 ad e rite ad ( 1 - r )" k = f V k ) x (d< k) = l, r ^ O ) X f j W =o No, differetiatig, e have xkrf > d r ' V M = f ^ d ^ k ) x = ' =i ad comparig the coefficiets leads to d(k> o d( k + 1 ) r k -i so that -i J f d f - > = r j j ( k l ) d < ^k+-) =i =o ad e the have d W = r (k + - 1)! /! (k - 1)!

15 1969] IN A RECURRING POWER SERIES 55 REFERENCES 1. L. M. Mile-Thomso, The Calculus of Fiite Differeces, Macmilla ad Co., Ltd., Lodo, 196, p G. Chrystal, Textbook of Algebra, Vol. II, Dover Publicatios, Ic., Ne York, 1961, p. 52. The author ishes to thak L. Carlitz ad V. E. Hoggatt, Jr. for their ecouragemet. *. [Cotiued from p. 4. ] 1. V. E. Hoggatt, J r., "Fiboacci Numbers ad Geeralized Biomial Coefficiets," Fiboacci Quarterly, 5 (1967), pp Dov Jarde, "The Product of Sequeces ith a Commo Liear Recursio Formula of Order 2," publ. i Recurrig Sequeces, Jerusalem, 1958, pp Origial paper appeared i Hebre i Riveo Lematematika, 3 (1949), pp ; 38, beig a joit paper ith Th B Motzki. 12. Dov Jarde, "Nullifyig Coefficiets," Scripta Mathematica, 19(1953), pp Eugee E. Kohlbecker, "O a Geeralizatio of Multiomial Coefficiets for Fiboacci Sequeces," Fiboacci Quarterly, 4 (1966), pp S. G. Mohaty, "Restricted Compositios," Fiboacci Quarterly, 5(1967), pp Roseaa F. Torretto ad J. Alle Fuchs, "Geeralized Biomial Coefficiets," J^bojia^clJ^ 2 (1964), pp Morga Ward, "A Calculus of Sequeces," Amer, J. Math., 58 (1936), pp Stephe K. Jerbic s "Fiboomial Coefficiets A Fe Summatio Properties," Master f s Thesis 8 Sa Jose State College,, Sa Jose* Calif., 1968.