SOME IDENTITIES BETWEEN BASIC HYPERGEOMETRIC SERIES DERIVING FROM A NEW BAILEY-TYPE TRANSFORMATION

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1 SOME IDENTITIES BETWEEN BASIC HYPERGEOMETRIC SERIES DERIVING FROM A NEW BAILEY-TYPE TRANSFORMATION JAMES MC LAUGHLIN AND PETER ZIMMER Abstrct We prove ew Biley-type trsformtio reltig WP- Biley pirs We the use this trsformtio to derive umber of ew 3- d 4-term trsformtio formule betwee bsic hypergeometric series 1 Itroductio Biley s trsform c be stted s follows: Lemm 1 Subject to suitle covergece coditios if β α r U r V +r d γ δ r U r V r+ the r0 α γ 0 β δ The proof follows by switchig the order of summtio see [] pges for exmple Biley set U 1 V 1 x δ y z; ; x; yz d used the -Guss sum 0 r 11 φ 1 b; c; c/ c/ c/b; c c/; to get tht 1 x y z; β x/y x/z; yz x x/yz; 0 y z; x/y x/z; 0 x α yz Dte: April Mthemtics Subject Clssifictio Primry: 33D15 Secodry:11B65 05A19 Key words d phrses Q-Series bsic hypergeometric series Biley chis Biley trsform WP-Biley pirs 1

2 JAMES MC LAUGHLIN AND PETER ZIMMER where α 0 1 d 13 β r0 α r ; r x; +r Here we re employig the usul ottios Let d be complex umbers with < 1 uless otherwise stted The 1 0 ; 0 : 1 ; : 1 j for N j0 1 ; ; ; 1 ; ; : 1 j j0 1 ; ; ; 1 ; A r φ s bsic hypergeometric series is defied by [ ] 1 r rφ s ; x b 1 b s 1 ; ; r ; 1 1/ s+1 r x ; b 1 ; b s ; 0 I moder ottio the pir of seueces α β ove re termed Biley pir reltive to x/ Slter i [10] d [11] subseuetly used this trsformtio of Biley to derive 130 idetities of the Rogers-Rmuj type The first mjor vritios i Biley s costruct t 1 pper to be due to Bressoud [5] Aother vritio ws give by Sigh i [9] All of these vritios were put i more forml settig by Adrews i [1] where he itroduced geerliztio of the stdrd Biley pir s defied t 13 see lso eutio 93 i [4] Defiitio Adrews [1] Two seueces α β form WP- Biley pir provided 14 β j0 / j +j j +j α j Note tht if 0 the the defiitio reverts to tht of stdrd Biley pir I the sme pper Adrews showed tht there were two distict wys to costruct ew WP-Biley pirs from give pir If α β stisfy 14 the so do α β d α β where α ρ 1 ρ 15 α c /ρ 1 /ρ c

3 SOME IDENTITIES DERIVING FROM A NEW BAILEY-TYPE TRANSFORMATION 3 β ρ 1/ ρ / /ρ 1 /ρ j0 1 c j ρ 1 ρ j /c j +j 1 cρ 1 / ρ / j c +j with c ρ 1 ρ / for the pir ove d α / 16 α β / j j j0 j β j j β j c c Adrews two costructios c be show to imply the followig Bileytype trsformtios for WP-Biley pirs ssumig suitle covergece coditios Theorem 1 If α β stisfy / j +j β α j j +j j0 the 1 ρ 1 ρ ; 17 1 /ρ 0 1 /ρ ; /ρ 1 ρ /ρ 1 /ρ ; /ρ 1 /ρ /ρ 1 ρ ; d 18 β 0 0 / /; / ; ρ 1 ρ β ρ 1 ρ ; /ρ 1 /ρ ; 0 ; /; ρ 1 ρ α α We hd iitilly derived the trsformtio t 17 i wy tht ws similr to the wy Biley derived 1 before fidig tht it followed from Adrews first costructio t 15 A result euivlet to the trsformtio t 18 ws lso stted by Bressoud i [5] I the preset pper we prove the followig trsformtio for WP-Biley pirs Theorem Subject to suitle covergece coditios if /; r ; +r 19 β α r ; r ; +r r0

4 4 JAMES MC LAUGHLIN AND PETER ZIMMER the / /b; / / ; 110 /; / b ; / /b / β /; 0 b b b ; + b 3 b 3 b ; b; b 0 b ; 0 b; 3 b 3 b ; α +1 α +1 We use this trsformtio to derive some ew 3- d 4-term trsformtios betwee bsic hypergeometric series A Trsformtio derivig from -log of Wtso s 3 F sum We recll the followig -logue of Wtso s 3 F sum see [6 II16 pge 355] [ ] λ λ λ b λ / λ / /λ 1 8φ 7 λ λ λ/ λ/b λ / ; λ We ow prove the mi theorem Proof of Theorem I Lemm 1 set U r /λ; r ; r V r λ; r λ /; r λ λ/; λ/ λ/b; b λ / b λ / ; λ / λ / b ; δ r λ λ b λ / λ /; r λ λ λ/ λ/b ; r The γ δ r U r V r+ r δ m+ U m V m+ m0 λ r

5 SOME IDENTITIES DERIVING FROM A NEW BAILEY-TYPE TRANSFORMATION 5 m0 λ λ b λ / λ /; m+ λ λ λ/ λ/b /λ; m ; m+ ; m λ λ; m+ λ /; m+ m+ λ λ b λ / λ /; λ λ λ/ λ/b λ; λ ; λ /; 1+ λ 1+ λ b λ / λ / ; m m0 λ λ λ 1+ / λ 1+ /b ; m λ /λ; m λ m λ 1+ / ; m λ λ b λ / λ /; λ λ λ/ λ/b ; λ; λ /; λ λ1+ λ/; λ 1+ / λ 1+ /b; 1+ b 1+ + λ / b + λ / ; 1+ λ + / λ / b ; λ λ/; λ/ λ/b 1+ b 1+ λ + / b λ + / ; λ / λ / b ; λ λ/; λ λ/ λ/b b λ / b λ / ; λ / λ / b ; b λ 3 / b λ 3 / ; λ / λ / b ; b; b; λ λ / b λ / ; eve b; 1 λ 3 / b λ 3 / ; odd 1 The fifth eulity comes from pplyig 1 to the sum from the lie before fter replcig λ with λ with d b with b i this idetity We ext me the substitutios λ /bc followed by c d gi suppose the seueces {α } d {β } re relted by β α r U r V +r r0 r0 /; r ; r ; +r ; +r α r The result ow follows We c use this theorem to re-derive some ow idetities betwee bsic hypergeometric series d lso to derive some ew idetities For exmple

6 6 JAMES MC LAUGHLIN AND PETER ZIMMER isertig the trivil WP-Biley pir α { > 0 β / ; ; i 110 leds to versio of 1 perhps ot surprisigly sice 1 ws used to prove Theorem We lso ote i pssig tht pplyig Adrews first costructio t 15 to this trivil WP-Biley pir leds to vrit of Jcso s sum of termitig 8 φ 7 while pplyig his secod costructio t 16 leds to vrit of the -Pfff-Slschütz sum Before goig further we itroduce some stdrd spce-svig ottio: [ 1 1 ] 1 4 r+1 r+1w r 1 ; 4 r+1 ; z r+1 φ r ; z Isertig the uit WP-Biley pir see [3] for exmple where this WP- Biley pir d others employed below my be foud α /; ; { 1 0 β 0 > 1 r+1 i 110 d replcig with leds to the followig idetity: b b ; ; b b b ; 8W 7 ; 1/ 1 1 / W 7 b; b b 3/ 3/ ; b ; b ; This idetity is prticulr cse of Biley s otermitig extesio of Jcso s 8 φ 7 sum see [6 pge 356 II5] We ow stte some trsformtios which we believe re ew Upo substitutig Sigh s WP-Biley pir [9] see lso [3] where oe of Adrews

7 SOME IDENTITIES DERIVING FROM A NEW BAILEY-TYPE TRANSFORMATION 7 costructios ws used to derive Sigh s pir from the uit pir α y z /yz; /y /z yz/; β y/ z/ /yz; /y /z yz/ ; ito 110 we get the followig trsformtio Corollry 1 3 / /b; / / ; /; 10 W 9 ; b y z yz ; b b b ; 1 W 11 ; b y y z z yz yz ; 1 1 y1 z1 /yz 1 1 /y1 /z1 yz/ b b 3 3 b ; 1 W 11 ; b y y z z yz 3 yz ; Remr: The idetity ove my be regrded s extesio of 1 sice substitutig y 1 i 3 leds to vrit of 1 We ext pply the theorem to some WP-Biley pirs foud by Adrews d Berovich [3] Corollry 4 / /b; / / ; /; 7 φ 6 b b ; 16 W 15 ; b b b ; b r r 1 1 /1 /1 / / s s ; A b b 3 3 b ;

8 8 JAMES MC LAUGHLIN AND PETER ZIMMER 0 16W ; s s b 3 3 Proof Isert the WP-Biley pir α /; /; /; ; β / ; ; s s ; A from [3] ito 110 Corollry 3 / /b; / / ; 5 /; [ 6 φ b ] 5 b ; b b b ; 0 r 16W ; b r s s b b 3 3 b ; 0 16W ; 1 ; A s s s s 1 b ; A Proof This trsformtio follows similrly fter isertig the WP-Biley pir ; α ; ; β ; from [3] ito 110 We ext cosider two WP-Biley pirs foud by Bressoud [5] see lso [3] where these pirs re lso ivestigted We do ot cosider Bressoud s first WP-Biley pir sice s remred i [3] it is limitig cse of Sigh s WP-Biley pir t

9 SOME IDENTITIES DERIVING FROM A NEW BAILEY-TYPE TRANSFORMATION 9 Corollry 4 6 / /b; / / ; /; 8 W 7 ; b 1 b b b ; ; ; ; 10 W 9 b b b b 3 3 b ; 10 W 9 ; b b 3 Proof Isert Bressoud s secod WP-Biley pir α 1 1 ; ; β ; ; ; ; ; ito 110 Corollry 5 / /b; / / ; 7 /; 8 W 7 ; b b b b ; ; ; 1W 11 i 1/4 i 1/4 b b b b 3 3 b ; r r 1W 11 ; i 3/ 1/4 i 3/ 1/4 p p b b ; ;

10 10 JAMES MC LAUGHLIN AND PETER ZIMMER Proof Isert Bressoud s third WP-Biley pir ito 110 α 1 1 ; ; ; β ; Filly we pply the theorem to three WP-Biley pirs foud by the preset uthors i [8]: α 1 / ; 8 β 1 / ; /; / 9 α / / /; / 1 ; / ; / β /; eve / 0 odd 10 α 3 d /d ; 1 /d d ; /d d/; / d /d; eve / β 3 /d d/; +1/ d /d; +1/ odd Note tht the third pir is restricted i the sese tht it is ecessry to set for 19 to hold Corollry 6 11 / /b; / / ; /; 8 W 7 ; b ; b b b ; 4φ 3 b ; b b

11 SOME IDENTITIES DERIVING FROM A NEW BAILEY-TYPE TRANSFORMATION b b 3 3 b ; 4φ 3 Proof Isert the WP-Biley pir t 8 ito 110 b 3 3 b 3 b 3 ; Remr: The substitutio lso gives specil cse of 1 Corollry 7 / /b; / / ; 1 /; 1W 11 ; b b 3 3 ; b b b ; 1W 11 ; b 3 3 ; b b 3 3 b ; 1W 11 ; b ; Proof Isert the WP-Biley pir t 9 ito 110 Corollry 8 b ; 3 ; 13 ; [ 14W 13 ; 3 b b 3 3 d d ; b1 1 d 1 d b 1 1 d 1 3 d 14W 13 3 ; 3 4 b b ] 5 3 d d ; b b 3 b ; 1W 11 ; b d d d d ; d 1 d b b d 1 d 3 b ;

12 1 JAMES MC LAUGHLIN AND PETER ZIMMER 1W 11 ; 3 b d d d 3 d ; Proof Isert the WP-Biley pir t 10 ito 110 d set Remr: The extr -products iserted i the umertors d deomitors of the terms i the two series o the left i the idetity ove re there so s to give ech these series the form of r+1 W r series Refereces [1] Adrews George E Biley s trsform lemm chis d tree Specil fuctios 000: curret perspective d future directios Tempe AZ 1 NATO Sci Ser II Mth Phys Chem 30 Kluwer Acd Publ Dordrecht 001 [] Adrews George E; Asey Richrd; Roy Rj Specil fuctios Ecyclopedi of Mthemtics d its Applictios 71 Cmbridge Uiversity Press Cmbridge 1999 xvi+664 pp [3] Adrews George; Berovich Alexder The WP-Biley tree d its implictios J Lodo Mth Soc o [4] Biley W N Some Idetities i Combitory Alysis Proc Lodo Mth Soc [5] Bressoud Dvid Some idetities for termitig -series Mth Proc Cmbridge Philos Soc o 11 3 [6] Gsper George; Rhm Miz Bsic hypergeometric series With foreword by Richrd Asey Secod editio Ecyclopedi of Mthemtics d its Applictios 96 Cmbridge Uiversity Press Cmbridge 004 xxvi+48 pp [7] Mc Lughli Jmes; Zimmer Peter Some Applictios of Biley-type Trsformtio - submitted [8] Mc Lughli Jmes; Zimmer Peter Some trsformtios for termitig bsic hypergeometric series - submitted [9] Sigh U B A ote o trsformtio of Biley Qurt J Mth Oxford Ser o [10] Slter L J A ew proof of Rogers s trsformtios of ifiite series Proc Lodo Mth Soc [11] Slter L J Further idetities of the Rogers-Rmuj type Proc Lodo MthSoc Mthemtics Deprtmet Aderso Hll West Chester Uiversity West Chester PA E-mil ddress: jmclughl@wcupedu Mthemtics Deprtmet Aderso Hll West Chester Uiversity West Chester PA E-mil ddress: pzimmer@wcupedu