Applicable Analysis and Discrete Mathematics available online at ABEL S METHOD ON SUMMATION BY PARTS.

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1 Applicable Analysis and Discrete Mathematics available online at Appl Anal Discrete Math 4 010), doi:1098/aadm c ABEL S METHOD ON SUMMATION BY PARTS AND BALANCED -SERIES IDENTITIES Wenchang Chu The Abel method on summation by parts is reformulated to present new and elementary proofs of several classical identities of terminating balanced basic hypergeometric series The examples strengthen our conviction that as traditional analytical instrument, the revised Abel method on summation by parts is indeed a very natural choice for woring with basic hypergeometric series 1 INTRODUCTION For an arbitrary complex seuence {τ }, define the bacward and forward difference operators and, respectively, by 1) τ = τ τ 1 and τ = τ τ +1 where is adopted for convenience in the present paper, which differs from the usual operator only in the minus sign Then Abel s lemma on summation by parts may be reformulated as B A = AB A 1 B 0 + A B =0 provided that the limit AB := lim A mb m+1 exists and one of the nonterminating series just displayed is convergent m In fact, according to the definition of the bacward difference, we have m B A = =0 m { } B A A 1 = =0 m =0 =0 A B m A 1 B 000 Mathematics Subject Classification 33D15, 05A30 Keywords and Phrases Abel s lemma on summation by parts, basic hypergeometric series =0 54

2 Abel s method on summation by parts and balanced -series identities 55 Replacing by 1 + for the last sum, we derive the following expression: m m { } B A = A m B m+1 A 1 B 0 + A B B +1 =0 = A m B m+1 A 1 B 0 + =0 m A B Letting m, we get the desired formula Recently, the author 4, 5 has systematically reviewed several fundamental basic hypergeometric series identities through the Abel lemma on summation by parts The approach can briefly be described as follows: Applying Abel s lemma on summation by parts to a given -series Ω, the machinery establishes a recurrence relation Iterating the recursive euation derives a transformation formula on the Ω- series involving a new free integer parameter m Truncating the Ω-series by specifying one of the parameters in the transformation yields a terminating series identity Finally, the limiting case m of the transformation if exists, of course) leads to a nonterminating series identity The objective of the present wor is to explore the applications of Abel s lemma on summation by parts to terminating balanced -series identities that, as a common underlying structure, involve -shifted factorials with bases and Several -series identities will be exemplified in a unified manner by means of Abel s lemma on summation by parts such as the -analogues of the second Gauss summation theorem and the Bailey formula on F 1 1/)-series due to Andrews 1; the terminating -analogues of the Watson and Whipple formulae on 3 F -series, discovered respectively by Andrews and Jain 8 Several new transformation formulae will also be established They will show again that as classical analytic weapon, Abel s lemma on summation by parts is indeed a very natural and powerful method in dealing with basic hypergeometric series identities In order to facilitate the readability of the paper, we reproduce the notations of -shifted factorial and basic hypergeometric series For two indeterminate and x, the shifted-factorial with base is defined by n 1 ) x; ) 0 = 1 and x; ) n = 1 x ) for n = 1,, The product and fraction of shifted factorials are abbreviated respectively to 3) 4) =0 =0 α, β,, γ; n = α; ) n β; ) n γ; ) n, α, β,, γ = α; ) n β; ) n γ; ) n A, B,, C A; ) n B; ) n C; ) n n

3 56 Wenchang Chu Following Bailey 3, Gasper-Rahman 6 and Slater 9, the basic hypergeometric series is defined by a0, a 5) 1,, a λ 1+λφ λ ; z b 1,, b λ = n=0 a0, a 1,, a λ z n, b 1,, b λ n where the base will be restricted to < 1 for nonterminating -series When a 0 a 1 a λ = b 1 b b λ, the series just defined is called balanced, which will be the main subject of the present paper -ANALOGUE OF GAUSS SECOND SUMMATION THEOREM Define the G- function by 6) Ga, b) : = 0 For the two seuences defined by A = a, b, ab a; ) b; ) ; ) ab; ) it is almost trivial to compute the limiting relation as well as the finite differences A = a, b, ab +1 and B = ab; ) ab; ) A 1 B 0 = AB = 0 and ) B = 1 ab; ) 1 ab 3 ab; ) Then we can manipulate the G-series by Abel s lemma on summation by parts as follows: Ga, b) = 0 a; ) b; ) ; ) ab; ) +1 = B A = A B 0 0 = 1 a; ) b; ) 1 ab ; ) 3 ab; ) 0 1 a)1 b) = 1 ab)1 3 ab) 0 ) a; ) b; ) ; ) 5 ab; ) +1 )

4 Abel s method on summation by parts and balanced -series identities 57 where the last passage has been justified by shifting the summation index 1+ Therefore, we have established the following recurrence relation 7) Ga, b) = G a, 1 a)1 b) b) 1 ab)1 3 ab) Iterating this relation m-times, we get the following functional euation Lemma 1 Recurrence relation on nonterminating series) Ga, b) = G m a, m b) a; ) m b; ) m ab; ) m Letting m and then appealing to Euler s -exponential function cf 6, II) n 0 n ) x) n ; ) n = x; ) we get the following limiting relation lim m Gm a, m b) = 0 +1 = ; ) = ; ) 1 ; ) which leads to the -analogue of Gauss second theorem cf Bailey 3, 4) Corollary Andrews 1, E 18) =0 a; ) b; ) ; ) ab; ) +1 = a; ) b; ) ; ) ab; ) 3 -ANALOGUE OF BAILEY S SUMMATION THEOREM Define the H- function by 8) Ha, c) : = 0 a; ) /a; ) ; ) c; ) ) c For the two seuences defined by A = /c; ) ; ) +1 c and B = 1/c, c

5 58 Wenchang Chu it is not hard to chec the limiting relation A 1 B 0 = AB = 0 as well as the finite differences A = 1 c 1/c; ) ) 1 c ; c and B = ) 1 ac)1 c/a) 1 c)1 c) /c, c Then we can reformulate the H-series by Abel s lemma on summation by parts as follows: Ha, c) = 0 a; ) /a; ) ; ) c; ) ) c = B A = A B ac)1 c/a) a; ) /a; ) = 1 c)1 c) ; ) c ) c; ) 0 Therefore, we have established the following recurrence relation 9) Ha, c) = Ha, c) 1 ac)1 c/a) 1 c)1 c) Iterating this relation m-times, we get the following functional euation Lemma 3 Recurrence relation on nonterminating series) Ha, c) = Ha, m c) ac; ) m c/a; ) m c; ) m Letting m and then noting that Ha, 0) = 1, we derive the following -analogue of Bailey s summation theorem cf Bailey 3, 4) Corollary 4 Andrews 1, E 19) =0 a; ) /a; ) ; ) c; ) ) c = ac; ) c/a; ) c; ) 4 -WATSON SUMMATION FORMULAE DUE TO ANDREWS AND JAIN Define the W- function by a, b, c, c 10) Wa, b, c) : = 4 φ 3 ; c, ab, ab

6 Abel s method on summation by parts and balanced -series identities 59 For the two seuences defined by a, b C =, ab it is not difficult to compute the limiting relation and D = ab; ) c; ) ab; ) c; ) C 1 D 0 = 0 and CD = c; ) ab; ) a, b, c as well as the finite differences C = a, b and D, ab = 1 )1 ab/c) ab; ) c; ) 1 ab)1 1/c) 3 ab; ) c; ) Then the W-series can be manipulated through Abel s lemma on summation by parts as follows: Wa, b, c) = 0 = c; ) a, b ab; ), c = c; ) a, b ab; ), c a; ) b; ) c; ) ; ) c; ) ab; ) ab/c) 1 ab)1 1/c) = D C = CD + C D ) a; ) b; ) c; ) ; ) c; ) 3 ab; ) 0 1 a)1 b)1 ab/c) 1 ab)1 3 ab)1 1/c) 0 a; ) b; ) c; ) ; ) c; ) 5 ab; ) where the last passage has been justified by shifting the summation index 1+ Therefore, we have established the following recurrence relation 11) Wa, b, c) = W a, b, 1 a)1 b)1 ab/c) c) 1 ab)1 3 ab)1 1/c) + c; ) a, b ab; ), c Iterating this relation m-times leads us to the following partial sum expression Wa, b, c) = W m a, m b, m c) a, b, ab/c; m ab; ) m c; m c) m ) m m 1 + =0 c) a, b, ab/c; ab; ) c; ) c; ) 1+4 ab; ) 1+ a, 1+ b, c which can further be simplified to the following transformation theorem Theorem 5 Transformation on balanced 4 φ 3 -series) Wa, b, c) = W m a, m b, m c) a, b, ab/c; m ab; ) m c; m c) m ) m + c; ) ab; ) a, b, c m 1 =0 ab/c a, c) b

7 60 Wenchang Chu Taing c = m in this theorem, we recover directly Jain s -analogue of Watson s summation formula cf Bailey 3, 33) Corollary 6 Jain s terminating 4 φ 3 -series identity 8, E 317 see also 7, E 15) n a; ) b; ) n ; ) ; ) ab; ) n = a; ) n b; ) n ; ) ; ) n ab; ) n =0 Instead, if we let b = δ m with δ = 0, 1 in Theorem 5, then we will obtain another terminating -analogue of the above mentioned Watson s summation formula Corollary 7 Andrews terminating 4 φ 3 -series identity, Thm 1; see also 7, E 14 and 10, E 11) For b = n with n being a nonnegative integer, there holds the following identity a, b, c, c 4φ 3 c, ab, ; ab, c/a, n even; = /a, c n/ 0, n odd In addition, the limiting case m of Theorem 5 yields the following surprising transformation formula on nonterminating series Proposition 8 Transformation on nonterminating balanced series) 4φ 3 a, b, c, c c, ab, ab ; = c; ) a, b ab; ), c =0 ab/c a, b c) 5 -WHIPPLE SUMMATION FORMULAE Define the M-function by 1) Ma, c, e) : = 4 φ 3, c, c, e, c/e For the two seuences defined by C = a; ) ac; ) c; ) ; ) and D = it is routine to verify the limiting relation C 1 D 0 = 0 and CD = c; ) ; ) ; 1/a, ac e, c/e a, 1/a e, c/e

8 Abel s method on summation by parts and balanced -series identities 61 as well as the finite differences C = 1 /a a; ) c; ) 1 1/a ac; ) ;, ) 1 ae)1 ac/e) 1/a, ac D = a1 e)1 c/e) e, c/e Then the M-series can be reformulated through Abel s lemma on summation by parts as follows: Ma, c, e) = a; ) /a; ) c; ) ; = D C = CD + C D ) e; ) c/e; ) = c; ) a, 1/a 1 ae)1 ac/e) a; ) 1/a; ) c; ) ; ) e, c/e a1 e)1 c/e) ; ) e; ) c/e; ) + 0 Therefore, we have established the following recurrence relation 13) Ma, c, e) = Ma, 1 ae)1 ac/e) c, e) a1 e)1 c/e) + c; ) ; ) a, 1/a e, c/e Iterating this relation m-times, we get the following partial sum expression Ma, c, e) = M m a, m c, m e) ae; ) m ac/e; ) m e; ) m c/e; ) m m 1 ) + a ae; ) ac/e; ) +1 a, /a e; ) c/e; ) e, +1 c/e =0 m a m + c; ) ; ) Simplifying the last partial sum, we derive the following transformation theorem Theorem 9 Transformation on balanced 4 φ 3 -series) Ma, c, e) = M m a, m c, m e) ae; ) m ac/e; m ) m a m e; ) m c/e; ) m + c; ) m 1 a, 1/a ; 1) ae, ac/e ) e, c/e a c =0 Taing a = m in this theorem, we recover directly the following terminating -analogue of Whipple s summation formula cf Bailey 3, 34) Corollary 10 Andrews terminating 4 φ 3 -series identity, Thm ; see also 10, E 1) For a = m with m being a nonnegative integer, there holds the following identity m+1 ; = 4φ 3, c, c, e, c/e ae; ) m ac/e; ) m e; ) m c/e; ) m

9 6 Wenchang Chu Instead, if we let c = m in Theorem 9, then we will obtain another terminating -analogue of the above mentioned Whipple s summation formula Corollary 11 Jain s terminating 4 φ 3 -series identity 7, E 319), 4φ m, m 3, e, 1 m ; /e = ae; ) m e/a; ) m e; ) m 6 TRANSFORMATION ON NONTERMINATING -WHIPPLE SUMS This section will further investigate transformations on the M-series defined in the last section It will finally be expressed in terms of φ 1 -series 61 For the two seuences defined by E = c; ) ; ) e/c; ) e; ) and F = c/e, 1 e/c we have no difficulty to confirm the limiting relation c; ) E 1 F 0 = 0 and EF = c/e e, c/e ; ) as well as the finite differences E = 1 +1 c/e 1 c/e F = 1 ac/e)1 c/ae) 1 c/e)1 c/e) c; ) 1 e/c; ) ;, ) e; ) 3 c/e, e/c Then Abel s lemma on summation by parts can be used to manipulate the M-series defined in the last section as follows: Ma, c, e) = a; ) /a; ) c; ) = F ; E = EF + E F ) e; ) c/e; ) = c c; ) + 1 ac/e)1 c/ae) a; ) /a; ) c; ) e e, c/e ; ) 1 c/e)1 c/e) ; ) e; ) 3 c/e; ) 0 Therefore, we have established another recurrence relation 14) Ma, c, e) = Ma, c, e) 1 ac/e)1 c/ae) 1 c/e)1 c/e) c e e, c/e c; ) ; )

10 Abel s method on summation by parts and balanced -series identities 63 Iterating this relation m-times, we get the following partial sum expression Ma, c, e) = Ma, m c, e) ac/e; ) m c/ae; ) m c/e; ) m m 1 c e =0 ac/e, c/ae c/e, c/e + c; ) ; ) e, 1+ c/e Simplifying the last partial sum, we derive the following transformation theorem Theorem 1 Transformation on balanced 4 φ 3 -series) Ma, c, e) = Ma, m c, e) ac/e; ) m c/ae; ) m c/e; ) m c c; ) m 1 ac/e, e ; c/ae ) e, c/e c Taing c = m in this theorem, we recover the terminating balanced series identity stated in Corollary 11 If letting m in Theorem 1, then we derive the following interesting transformation Corollary 13 Nonterminating transformation formula), c, c 0, ac/e; ) c/ae; ) 4φ 3 ; =, e, c/e 3 φ ;, e c/e; ) c c; ) ac/e, e ; c/ae ) e, c/e c 6 Denote the last 3 φ -series by =0 0, 15) Sa, e) : = 3 φ, e For the two seuences defined by E = /e; ) ; ) and F = =0 ; 1/e, e we can compute without difficulty the limiting relation E 1 F 0 = 0 and EF = as well as the finite differences e ; ) e E = 1 e 1/e; ) 1 e ; 1 ae)1 e/a) and F = ) 1 e)1 e) /e, e

11 64 Wenchang Chu Then Abel s lemma on summation by parts can be utilized to reformulate the S- series as follows: Sa, e) = a; ) /a; ) ; = F E = EF + E F ) e; ) = e 1 ae)1 e/a) a; ) /a; ) ; + ) e 1 e)1 e) ; ) e; ) Therefore, we have established the following recurrence relation 16) Sa, e) = Sa, 1 ae)1 e/a) e e) 1 e)1 e) ; ) e Iterating this relation m-times, we get the following partial sum expression Sa, e) = Sa, m e) ae; ) me/a; ) m e; ) m m 1 =0 0 ae; ) e/a; ) e; ) e e ; ) Simplifying the last partial sum, we derive the following transformation theorem Theorem 14 Transformation on nonterminating 3 φ -series) Sa, e) = Sa, m e) ae; ) me/a; ) m e; ) m e ; ) e m 1 =0 ae, e/a; Letting m in this theorem, we find the following very strange transformation Corollary 15 Transformation on nonterminating 3 φ -series) 0, 3φ ; =, e φ 1 ; ae; ) e/a; ) e; ) e ; ae, e/a; ) e =0 Substituting this transformation into Corollary 13, we derive the following less elegant transformation formula Corollary 16 Transformation formula on nonterminating balanced 4 φ 3 -series), c, c 4φ 3, e, c/e e ; ac/e, c/ae c c; ) e ; ) e, c/e ae, e/a, ac/e, c/ae; = φ 1 ; e, c/e; ae, e/a; e, c/e =0 ac/e, c/ae c =0

12 Abel s method on summation by parts and balanced -series identities 65 Taing a = m in this Corollary and then applying the -Chu-Vandermonde-Gauss summation theorem 6, II6, we recover the identity stated in Corollary 10 REFERENCES 1 G E Andrews: On the -analog of Kummer s theorem and applications Due Math J, ), G E Andrews: On -analogues of the Watson and Whipple summations SIAM J Math Anal, 7 3) 1976), W N Bailey: Generalized Hypergeometric Series Cambridge University Press, Cambridge, W Chu: Bailey s very well poised 6ψ 6-series identity Journal of Combinatorial Theory Series A), 113 6) 006), W Chu: Abel s lemma on summation by parts and basic hypergeometric series Advances in Applied Mathematics, 39 4) 007), G Gasper, M Rahman: Basic Hypergeometric Series nd ed) Cambridge University Press, Cambridge, Victor J W Guo: Elementary proofs of some -identities of Jacson and Andrews- Jain Discrete Math, ), V K Jain: Some transformations of basic hypergeometric functions II SIAM J Math Anal, 1 6) 1981), L J Slater: Generalized Hypergeometric Functions Cambridge University Press, Cambridge, A Verma, V K Jain: Some summation formulae for nonterminating basic hypergeometric series SIAM J Math Anal, 16 3) 1985), Dipartimento di Matematica, Received February, 009) Università del Salento, Revised October 7, 009) Lecce-Arnesano P O Box 193, Lecce, Italy chuwenchang@unileit