Two contiguous relations of Carlitz and Willett for balanced series Wenchang Chu and Xiaoyuan Wang

Transcription

1 Lecture Notes of Seminario Interdisciplinare di Matematica Vol 9(200), pp Two contiguous relations of Carlitz and Willett for balanced series Wenchang Chu and Xiaoyuan Wang Abstract The modified Abel lemma on summation by parts is employed to review two unusual contiguous relations discovered by Willett (967) and Carlitz (969), respectively, for classical series and basic 4 3 series The modified Abel lemma on summation by parts For an arbitrary complex sequence { }, define the bacward and forward difference operators r and,respectively,by r and where is adopted for convenience in the present paper, which di ers from the usual operator only in the minus sign Then Abel s lemma on summation by parts may be reformulated as X X B ra [AB] A B 0 A B 0 provided that the limit [AB] : lim m! A m B m exists and one of the nonterminating series just displayed is convergent In fact, according to the definition of the bacward di erence, we have X m B ra B {A A } A B A B 0 0 Replacing by for the last sum, we derive the following expression: B ra A m B m A B 0 A {B B } 0 A m B m A B A B Authors address: W Chu, Università degli Studi di Salento, Dipartimento di Matematica Ennio De Giorgi, Via provinciale Lecce-Arnesano, PO Box 93, 7300 Lecce, Italy; chuwenchang@unileit X Wang, Dalian Jiaotong University, School of Science, Dalian 6028, PR China; xiaoyuandlut@yahoocomcn Keywords Abel s lemma on summation by parts, classical hypergeometric series, basic hypergeometric series, balanced series, contiguous relation AMS Subject Classification 33C20, 33D5 0

2 26 Wenchang Chu and Xiaoyuan Wang Letting m!, we confirm the modified Abel lemma on summation by parts Recently, several important summation formulae of classical and basic hypergeometric series have been reviewed by Chu [3, 4] through the modified Abel lemma on summation by parts The same approach has further been employed by Chu and Wang [5] to derive numerous contiguous relations of classical 3 F 2 series There exist two very unusual relations on balanced series in literature One is discovered by Willett [7, 967] for the classical hypergeometric series Another one is its q analogue due to Carlitz [2, 969] However, their nonterminating forms derived by Carlitz [2] contain lacunae The objective of the present paper is to establish two nonterminating contiguous relations via the modified Abel lemma on summation by parts, that correct the mistaes appeared in Carlitz paper 2 Willett s contiguous relation For a complex x and a nonnegative integer n, define the shifted factorial by (x) 0 and (x) n (x n)/ (x) x(x ) (x n ) for n 2 N with its multiparameter forms being abbreviated to [,,, ] n ( ) n ( ) n ( ) n,,,, ( ) n ( ) n ( ) n A, B,,C (A) n (B) n (C) n n From the study of Fuchsian system of two first order linear di erential equations with three regular singular points, Willett [7] (see Carlitz [2, equation 2] also) discovered the following terminating hypergeometric series identity Proposition (Willett [7], equation 3) a, b, d n, n a n, b n, d n(n )(a d)(b d) ( a n)( b n)d(d ) 4 F 3 a, b, d n, 2 n 2a n, 2b n, 2d Here and forth, according to Bailey [, 2], the generalized hypergeometric series reads as a0,a,,a r X (a 0 ) n (a ) n (a r ) n rf s z z n c,,c s n!(c ) n (c s ) n Willett [7, equation 29], (see Carlitz [2, equation ] also) noted also that the last contiguous relation corresponds to the following equivalent products of two Gaussian 2 F series: 2F a, b d x 2F a, b d x n0 ab(a d)(b d) d 2 (d )(d ) x2 2 F a, b 2d x 2F a, b 2 d x

3 Two contiguous relations of Carlitz and Willett for balanced series 27 Carlitz [2, equation 3], pointed out that there holds the following nonterminating form a, b, c, c d a c, b c, d (a d)(b d)c( c) ( a c)( b c)d( d) 4 F 3 a, b, 2c, c d 2a c, 2b c, 2d Unfortunately, this result is false, which can be checed by letting b d and then applying the Gauss summation formula for 2 F series (cf Bailey [, 3]) a, c 2F a c ( a c) ( a) ( c) 60 By means of Abel s lemma on summation by parts, we are going to establish the correct nonterminating version of Willett s identity Define the two sequences by A a, c, a c and B b, c d d, b c It is not hard to chec the finite di erences a, c ra and B, a c ( c)(d b) d( b c) b, c d d, 2b c as well as the limiting relations A B 0 0 and [AB] ( a c) ( b c) (d) ( a) (b) ( c) ( c d) where the last equality is justified by A m B m b( c d) d( b c) b( c d) d( b c) a, b, c, 2c d, d, a c, 2b c ( a c) (2 b c) ( d) ( a) ( b) ( c) (2 c d) m ( a m) ( b m) ( c m) (2 c d m) ( m) ( d m) ( a c m) (2 b c m) because the last line tends to one as m!in view of the following asymptotic formula (cf Rainville [6, Lemma 7 in 8]) () (x m) (m )m x

4 28 Wenchang Chu and Xiaoyuan Wang According to the modified Abel lemma on summation by parts, we can manipulate the following balanced 4 F 3 series: a, b, c, c d a c, b c, d X B ra [AB] A B 0 X A B 0 0 ( a c) ( b c) (d) ( a) (b) ( c) ( c d) ( c)(d b) d( b c) X a, b, c, c d 0, a c, 2b c, d In terms of hypergeometric series, this can be expressed as the contiguous relation a, b, c, c d (2a) a c, b c, d ( a c) ( b c) (d) ( a) (b) ( c) ( c d) ( c)(d b) a, b, c, c d (2b) d( b c) a c, 2b c, d Alternatively, for the two sequences defined by b, 2c a, c d C and D, 2b c d, a c it is almost routine to calculate the finite di erences b, c c(d a) a, c d rc and D, 2b c ( a c)( d) 2d, 2a c as well as the limiting relations C D 0 0 and [CD] ( a c) (2 b c) ( d) ( a) ( b) (2 c) ( c d) where the last equality is justified by ( a)( c d) 2a, b, 2c, 2c d C m D m ( d)( a c), 2d, 2a c, 2b c ( a)( c d) ( d)( a c) (2 a c) (2 b c) (2 d) (2 a) ( b) (2 c) (2 c d) (2 a m) ( b m) (2 c m) (2 c d m) ( m) (2 d m) (2 a c m) (2 b c m) because the last line tends to one as m!thans again to the formula () By means of the modified Abel lemma on summation by parts, we can similarly reformulate balanced series as follows: a, b, c, c d a c, 2b c, d X D rc [CD] C D 0 X C D 0 0 m

5 Two contiguous relations of Carlitz and Willett for balanced series 29 ( a c) (2 b c) ( d) ( a) ( b) (2 c) ( c d) c(d a) X a, b, c d, 2c ( a c)( d), 2a c, 2b c, 2d 0 This can be stated as another contiguous relation a, b, c, c d (3a) 4 F 3 a c, 2b c, d ( a c) (2 b c) ( d) ( a) ( b) (2 c) ( c d) c(d a) a, b, 2c, c d (3b) ( a c)( d) 4 F 3 2a c, 2b c, 2d Substituting (3a)-(3b) into (2a)-(2b) and then simplifying the resulting equation, we get the following transformation formula Theorem 2 (Contiguous relation of 4 F 3 series) a, b, c, c d a c, b c, d ( a c) ( b c) ( d) ( a) ( b) ( c) ( c d) c( c)(a d)(b d) a, b, 2c, c d ( a c)( b c)d( d) 4 F 3 2a c, 2b c, 2d It is obvious that this theorem corrects the corresponding formula obtained by Carlitz [2, equation 3], where the term containing functions has been missing When the series are terminated by c n with n 2 N, this theorem reduces clearly to Willett s contiguous relation for terminating 4 F 3 series displayed in Proposition 3 Carlitz contiguous relation In comparison with the ordinary hypergeometric series, Carlitz [2, equation 8] derived a nonterminating q analogue of Willett s identity, which is unfortunately false again We need to reproduce the notations of q shifted factorial and basic hypergeometric series For two indeterminate x and q, the shifted-factorial of x with base q is defined by (x; q) 0 and (x; q) n ( x)( xq) ( xq n ) for n 2 N When q <, we have two well-defined infinite products Y (x; q) ( q x) and (x; q) n (x; q) /(xq n ; q) 0 The product and fraction of shifted factorials are abbreviated compactly to [,,, ; q] n ( ; q) n ( ; q) n ( ; q) n,,,, A, B,,C q n ( ; q) n( ; q) n ( ; q) n (A; q) n (B; q) n (C; q) n

6 30 Wenchang Chu and Xiaoyuan Wang Following Bailey [], the basic hypergeometric series is defined by a0,a,,a r X (a 0 ; q) n (a ; q) n (a r ; q) n r s q; z c,,c s (q; q) n (c ; q) n (c s ; q) n where the base q will be restricted to q < for nonterminating q series Then the following q analogue of Theorem 2 will be established n0 Theorem 3 (Contiguous relation of 4 3 series) a, b, c, qcd qa, qb, qc, qcd 4 3 qac, qbc, d q; q q, qac, qbc, qd q q( c)( qc)(a d)(b d) ( qac)( qbc)( d)( qd) 4 3 q 2 ac, q 2 bc, q 2 d q; q We remar that the formula just displayed corrects the corresponding formula obtained by Carlitz [2, equation 8], where the quotient of infinite products has been missing This theorem can analogously be proved through the modified Abel lemma on summation by parts Proof For the two sequences given by qa, qc A q, qac q and B b, qcd d, qbc q it is not di cult to verify the finite di erences a, c ra q, qac q q (b d)( qc) and B ( d)( qbc) as well as the limiting relations A B 0 0 and [AB] qa, b, qc, qcd q, qac, qbc, d q z n b, qcd qd, q 2 bc q In view of the modified Abel lemma on summation by parts, we can reformulate the following balanced 4 3 series a, b, c, qcd 4 3 qac, qbc, d q; q X B ra [AB] A B 0 X A B 0 0 qa, b, qc, qcd q, qac, qbc, d q (b d)( qc) X qa, b, qc, qcd ( d)( qbc) q, qac, q 2 bc, qd q q 0 which leads us to the following contiguous relation a, b, c, qcd qa, b, qc, qcd (4a) 4 3 qac, qbc, d q; q q, qac, qbc, d q (b d)( qc) (4b) ( d)( qbc) 4 3 qa, b, qc, qcd qac, q 2 bc, qd q; q q

7 Two contiguous relations of Carlitz and Willett for balanced series 3 Alternatively, with the two sequences defined by qb, q 2 c qa, qcd C q, q 2 bc q and D qd, qac q we can compute the finite di erences b, qc rc q, q 2 bc q q q( c)(a d) and D ( qac)( qd) as well as the limiting relations C D 0 0 and [CD] q, qac, q 2 bc, qd q qa, qcd q 2 d, q 2 ac q Applying the modified Abel lemma on summation by parts, we can similarly manipulate the balanced 4 3 series as follows: qa, b, qc, qcd 4 3 qac, q 2 bc, qd q; q X D rc [CD] C D 0 X C D 0 0 q, qac, q 2 bc, qd q q( c)(a d) ( qac)( qd) X 0 This can be stated as another contiguous relation qa, b, qc, qcd qa, qb, q 2 (5a) 4 3 qac, q 2 bc, qd q; q c, qcd q, qac, q 2 bc, qd q q( c)(a d) (5b) ( qac)( qd) 4 3 q, q 2 ac, q 2 bc, q 2 d q q 2 ac, q 2 bc, q 2 d q; q Substituting (5a)-(5b) into (4a)-(4b) and then simplifying the resulting equation, we get the transformation formula displayed in Theorem 3 When c q n with n 2 N, Theorem 3 reduces to the following contiguous relation for terminating 4 3 series, which is attributed to Carlitz [2] even though his formula contains some slight misprints Proposition 4 (Carlitz [2], equation : n 2 N) a, b, q n d, q n 4 3 q n a, q n b, d q; q (a d)(b d)( qn )( q n ) ( d)( qd)(a q n )(b q n ) qa, qb, q n d, q 2 n 4 3 q 2 n a, q 2 n b, q 2 d q; q As shown by Carlitz [2], this contiguous relation can alternatively be expressed as the following equivalent products of two 2 series q q

8 32 Wenchang Chu and Xiaoyuan Wang Corollary 5 (Carlitz [2], equation 5) a, b 2 q; dx d ab 2 a,b d q; x ( a )( b )( da )( db ) ( d)( qd)( d )( qd x 2 ) qa, qb 2 q; dx qa,qb q 2 2 q; x d ab q 2 d References [] W N Bailey, Generalized hypergeometric series, CambridgeUniversityPress, Cambridge, 935 [2] L Carlitz, A transformation formula for an 4 F 3, Quart J Math Oxford Ser (2), 20(969), 7 20 [3] W Chu, Abel s method on summation by parts and hypergeometric series, J Di erence Equ Appl, 2(8)(2006), [4] W Chu, Abel s method on summation by parts and basic hypergeometric series, Adv Appl Math, 39(4)(2007), [5] W Chu & XX Wang, Abel s method on summation by parts and hypergeometric contiguous relations, IntegralTransformsSpecFunct,8()(2007), [6] ED Rainville, Special functions, ChelseaPublishingCompany, Bronx, NewYor, 97 [7] RA Willett, A new identity between certain products of hypergeometric functions, Quart J Math Oxford Ser (2), 8(967),