J. Math. Anal. Appl. Some identities between basic hypergeometric series deriving from a new Bailey-type transformation

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1 J. Mth. Anl. Appl Contents lists ville t ScienceDirect J. Mth. Anl. Appl. Some identities between bsic hypergeometric series deriving from new Biley-type trnsformtion Jmes Mc Lughlin Peter Zimmer Mthemtics Deprtment Anderson Hll West Chester University West Chester PA USA rticle info strct Article history: Received 4 Jnury 008 Aville online 4 April 008 Submitted by B.C. Berndt Keywords: Q-series Bsic hypergeometric series Biley chins Biley trnsform WP-Biley pirs We prove new Biley-type trnsformtion relting WP-Biley pirs. We then use this trnsformtion to derive number of new 3- nd 4-term trnsformtion formule between bsic hypergeometric series. 008 Elsevier Inc. All rights reserved. 1. Introduction Biley s trnsform cn be stted s follows: Lemm 1. Subject to suitle convergence conditions if then β n α r U n r V n+r nd γ n δ r U r n V r+n r0 rn α n γ n β n δ n. n0 n0 The proof follows by switching the order of summtion see [ pp ] for exmple. Biley set U n 1 ; n V n 1 x; n nd used the -Guss sum n x δ n y z; n yz φ 1 b; c; c/ c/ c/b; c c/; 1.1 * Corresponding uthor. E-mil ddresses: jmclughl@wcup.edu J. Mc Lughlin pzimmer@wcup.edu P. Zimmer X/$ see front mtter 008 Elsevier Inc. All rights reserved. doi: /j.jm

2 J. Mc Lughlin P. Zimmer / J. Mth. Anl. Appl to get tht n x y z; n β n x/y x/z; n y z; n x α n 1. yz x x/yz; x/y x/z; n yz n0 where α 0 1 nd β n r0 α r ; n r x; n+r. n0 Here we re employing the usul nottions. Let nd be complex numbers with < 1 unless otherwise stted. Then n 1 0 ; 0 : 1 n ; n : 1 j for n N j0 1 ; n ; n... ; n 1... ; n ; : 1 j j0 1 ; ;... ; 1... ;. An r φ s bsic hypergeometric series is defined by [ ] 1... r 1 ; n ; n... r ; n rφ s ; x 1 n nn 1/ s+1 r x n. b 1...b s ; n b 1 ; n...b s ; n n0 In modern nottion the pir of seuences α n β n ove re termed Biley pir reltive to x/. Slter in [910] subseuently used this trnsformtion of Biley to derive 130 identities of the Rogers Rmnujn type. The first mjor vritions in Biley s construct t 1. pper to be due to Bressoud [5]. Another vrition ws given by Singh in [8]. All of these vritions were put in more forml setting by Andrews in [1] where he introduced generliztion of the stndrd Biley pir s defined t 1.3 see lso E. 9.3 in [4]. Definition. See Andrews [1]. Two seuences α n β n form WP-Biley pir provided 1.3 β n j0 / n j n+ j n j n+ j α j. Note tht if 0 then the definition reverts to tht of stndrd Biley pir. 1.4 In the sme pper Andrews showed tht there were two distinct wys to construct new WP-Biley pirs from given pir. If α n β n stisfy 1.4 then so do α n β n nd α n β n where n α n c α ρ n 1 ρ n /ρ 1 /ρ n β n ρ 1/ ρ / n /ρ 1 /ρ n c 1 c j ρ 1 ρ j /c n j n+ j 1 cρ 1 / ρ / j n j c n+ j j0 j β j c 1.5 with c ρ 1 ρ / for the pir ove nd α n / n n α n n β / n j j n β n j j. 1.6 j0 Andrews two constructions cn be shown to imply the following Biley-type trnsformtions for WP-Biley pirs ssuming suitle convergence conditions. c Theorem 1. If α n β n stisfy β n j0 / n j n+ j n j n+ j α j

3 67 J. Mc Lughlin P. Zimmer / J. Mth. Anl. Appl then nd n0 n0 1 n ρ 1 ρ ; n 1 /ρ 1 /ρ ; n ρ 1 ρ /ρ 1ρ /ρ 1 /ρ ; /ρ 1 /ρ /ρ 1 ρ ; n β n ρ 1 ρ n ; n α n 1.7 /ρ 1 /ρ ; n ρ 1 ρ n0 n β n / /; ; n n α / ; /; n n. 1.8 n0 We hd initilly derived the trnsformtion t 1.7 in wy tht ws similr to the wy Biley derived 1. before finding tht it followed from Andrews first construction t 1.5. A result euivlent to the trnsformtion t 1.8 ws lso stted by Bressoud in [5]. In the present pper we prove the following trnsformtion for WP-Biley pirs. Theorem. Subject to suitle convergence conditions if β n r0 /; n r ; n r ; n+r ; n+r α r 1.9 then / /b; / / ; /; b b b ; n0 + b 3 b 3 b ; n0 n0 b; n b b ; n b; n 3 b 3 b ; n / b ; n / /b / /; n n α n n β n n+1 α n We use this trnsformtion to derive some new 3- nd 4-term trnsformtions between bsic hypergeometric series.. A trnsformtion deriving from -nlog of Wtson s 3 F sum We recll the following -nlogue of Wtson s 3 F sum see [6 II.16 p. 355] [ ] λ λ λ bλ / λ / /λ 8φ 7 λ λ λ/λ/bλ / λ ; λλ/; b λ / b λ / ;. λ/λ/b; λ / λ / b ; We now prove the min theorem..1 Proof of Theorem. In Lemm 1 set Then U r /λ; r ; r V r λ; r λ /; r δ r λ λ bλ / λ /; r λ λ λ/λ/b ; r r λ.

4 J. Mc Lughlin P. Zimmer / J. Mth. Anl. Appl γ n δ r U r n V r+n rn δ m+n U m V m+n m0 λ λ bλ / λ m+n /; m+n λ λ λ/λ/b /λ; m λ; m+n λ ; m0 m+n ; m λ /; m+n λ λ bλ / λ n /; n λ λ λ/λ/b λ; n λ ; n λ /; n 1+n λ 1+n λ n b n λ / n λ / n ; m λ m0 n λ n λ 1+n /λ 1+n /b n λ n /λ; m λ n ; m λ 1+n / ; m λ λ bλ / λ n /; n λ λ λ/λ/b λ; n λ ; n λ /; n λ1+n λ/; 1+n b 1+n +n λ / b +n λ / ; λ 1+n /λ 1+n /b; 1+n λ +n / λ / b ; λλ/; 1+n b 1+n λ +n / bλ +n / ; n λ b; λ/λ/b λ /λ / b ; n λλ/; n λ λ/λ/b λ / bλ / ; λ /λ / b ; λ 3 / bλ 3 / ; λ /λ / b ; ; n n even λ / bλ / ; n ; n 1 n odd. λ 3 / bλ 3 / ; n 1 The fifth eulity comes from pplying.1 to the sum from the line before fter replcing λ with λ n with n nd b with b n in this identity. We next me the substitutions λ /bc followed by c nd gin suppose the seuences {α n } nd {β n } re relted by β n α r U n r V n+r r0 The result now follows. r0 /; n r ; n r ; n+r ; n+r α r. We cn use this theorem to re-derive some nown identities between bsic hypergeometric series nd lso to derive some new identities. For exmple inserting the trivil WP-Biley pir { 1 n 0 α n 0 n > 0 β n / ; n ; n in 1.10 leds to version of.1 perhps not surprisingly since.1 ws used to prove Theorem. We lso note in pssing tht pplying Andrews first construction t 1.5 to this trivil WP-Biley pir leds to vrint of Jcson s sum of terminting 8 φ 7 while pplying his second construction t 1.6 leds to vrint of the -Pfff Slschütz sum. Before going further we introduce some stndrd spce-sving nottion: [ 1 1 ] r+1 r+1w r 1 ; 4... r+1 ; z r+1 φ r ; z. 1 r+1 Inserting the unit WP-Biley pir see [3] for exmple where this WP-Biley pir nd others employed below my be found α n n /; n ; n { 1 n 0 β n 0 n > 1 in 1.10 nd replcing with leds to the following identity: m

5 674 J. Mc Lughlin P. Zimmer / J. Mth. Anl. Appl b b ; b b b ; 1/ 1 1 / 1 1 ; 8W 7 ; b b 3/ 3/ ; b b; 8W 7 ; b ;. This identity is prticulr cse of Biley s nonterminting extension of Jcson s 8 φ 7 sum see [6 II.5 p. 356]. We now stte some trnsformtions which we believe re new. Upon substituting Singh s WP-Biley pir [8] see lso [3] where one of Andrews constructions ws used to derive Singh s pir from the unit pir α n y z n /yz; n /y /z yz/; n β n y/ z/ /yz; n /y /z yz/ ; n into 1.10 we get the following trnsformtion.. Corollry 1. / /b; / / ; 10 W 9 ; /; b y z yz b b b ; 1W 11 ; b y y z z yz yz ; 1 1 y1 z1 /yz 1 1 /y1 /z1 yz/ b ; b b W 11 ; b y y z z yz 3 yz ; ;..3 Remr. The identity ove my be regrded s n extension of.1 since substituting y 1 in.3 leds to vrint of.1. We next pply the theorem to some WP-Biley pirs found by Andrews nd Berovich [3]. Corollry. / /b; / / ; 7φ 6 b /; b b b b ; 16 W 15 ; b 1 1 /1 /1 / / 16 W 15 ; b 3 3 Proof. Insert the WP-Biley pir α n /; n n /; n /; n ; n β n / ; n ; n from [3] into b b 3 3 ; 3 ; b ; ;..4

6 J. Mc Lughlin P. Zimmer / J. Mth. Anl. Appl Corollry 3. / /b; / / ; 6φ 5 b /; b b b b ; ; 16 W 15 ; b 3 3 ; b b 3 3 b ; 16 W 15 ; b Proof. This trnsformtion follows similrly fter inserting the WP-Biley pir α n ; n n ; n β n ; n ; n from [3] into ;..5 We next consider two WP-Biley pirs found by Bressoud [5] see lso [3] where these pirs re lso investigted. We do not consider Bressoud s first WP-Biley pir since s remred in [3] it is limiting cse of Singh s WP-Biley pir t.. Corollry 4. / /b; / / ; 8W 7 ; /; b ; b b ; b ; 10 W 9 b b ; b b 3 3 b ; 10 W 9 ; b b 3. ;.6 Proof. Insert Bressoud s second WP-Biley pir into α n 1 n 1 ; n n ; n β n ; n ; n ; n ; n n Corollry 5. / /b; / / ; 8W 7 ; /; b ; b b ; b ; 1W 11 i 1/4 i 1/4 b b ;

7 676 J. Mc Lughlin P. Zimmer / J. Mth. Anl. Appl b b 3 3 b ; 1 W 11 ; i 3/ 1/4 i 3/ 1/4 b b ;..7 Proof. Insert Bressoud s third WP-Biley pir α n 1 n ; n n 1 ; n into β n ; n ; n n Finlly we pply the theorem to three WP-Biley pirs found by the present uthors in [7]: α 1 n / n ; n n β 1 n / ; n /; n / n n.8 α n / / /; n / 1 n ; n β n { / ; n/ /; n/ n even 0 n odd α 3 n d /d ; n 1 n /d d ; n n β 3 /dd/; n/ d /d; n/ /dd/; n+1/ d/d; n+1/ n even n odd Note tht the third pir is restricted in the sense tht it is necessry to set for 1.9 to hold. Corollry 6. / /b; / / ; 8W 7 ; b /; ; b b [ b ; b ] 4φ 3 ; b b 1 1 b b 3 3 b ; [ 4φ b b 3 b 3 ] ;..11 Proof. Insert the WP-Biley pir t.8 into Remr. The substitution lso gives specil cse of.1. Corollry 7. / /b; / / ; /; 1W 11 b b b ; ; 1W 11 ; b b b ; ;

8 J. Mc Lughlin P. Zimmer / J. Mth. Anl. Appl b b W 11 ; b b ; ;..1 Proof. Insert the WP-Biley pir t.9 into Corollry 8. b ; 3 ; ; b1 1 d 1 d d1 b 1 d 14 W 13 3 ; 3 b b 3 b ; 4 b b [14 W 13 ; 3 b b 3 3 d d ; 3 1W 11 ; ] d d ; b d d d d ; b b 3 b ; d1 d d d 1 W 11 ; 3 b d d d 3 d ;..13 Proof. Insert the WP-Biley pir t.10 into 1.10 nd set. Remr. The extr -products inserted in the numertors nd denomintors of the terms in the two series on the left in the identity ove re there so s to give ech these series the form of n r+1 W r series. References [1] George E. Andrews Biley s trnsform lemm chins nd tree in: Specil Functions 000: Current Perspective nd Future Directions Tempe AZ in: NATO Sci. Ser. II Mth. Phys. Chem. vol. 30 Kluwer Acd. Publ. Dordrecht 001 pp. 1. [] George E. Andrews Richrd Asey Rnjn Roy Specil Functions Encyclopedi Mth. Appl. vol. 71 Cmbridge Univ. Press Cmbridge 1999 xvi+664 pp. [3] George Andrews Alexnder Berovich The WP-Biley tree nd its implictions J. London Mth. Soc [4] W.N. Biley Some identities in combintory nlysis Proc. London Mth. Soc [5] Dvid Bressoud Some identities for terminting -series Mth. Proc. Cmbridge Philos. Soc [6] George Gsper Mizn Rhmn Bsic Hypergeometric Series with foreword by Richrd Asey second ed. Encyclopedi Mth. Appl. vol. 96 Cmbridge Univ. Press Cmbridge 004 xxvi+48 pp. [7] Jmes Mc Lughlin Peter Zimmer Some trnsformtions for terminting bsic hypergeometric series submitted for publiction. [8] U.B. Singh A note on trnsformtion of Biley Q. J. Mth. Oxford Ser [9] L.J. Slter A new proof of Rogers s trnsformtions of infinite series Proc. London Mth. Soc [10] L.J. Slter Further identities of the Rogers Rmnujn type Proc. London Mth. Soc