Some implications of Chu s 10 ψ 10 extension of Bailey s 6 ψ 6 summation formula

Transcription

1 Some implicatios of Chu s 10 ψ 10 extesio of Bailey s 6 ψ 6 summatio fomula James McLaughli Adew V Sills Pete Zimme August Keywods: q-seies Roges-Ramauja Type Idetities Bailey chais False Theta Seies Subject Class: [000]Pimay: 33D15 Secoday:11B65 05A19 Abstact Lucy Slate used Bailey s 6 ψ 6 summatio fomula to deive the Bailey pais she used to costuct he famous list of 130 idetities of the Roges-Ramauja type I the peset pape we apply the same techiques to Chu s 10ψ 10 geealizatio of Bailey s fomula to poduce quite geeal Bailey pais Slate s Bailey pais ae the ecoveed as special limitig cases of these moe geeal pais I e-examiig Slate s wok we fid that he Bailey pais ae fo the most pat special cases of moe geeal Bailey pais cotaiig oe o moe fee paametes Futhe we also fid ew geeal Bailey pais (cotaiig oe o moe fee paametes) which ae also implied by the 6ψ 6 summatio fomula Slate used the Jacobi tiple poduct idetity (sometimes coupled with the quituple poduct idetity) to deive he ifiite poducts Hee we also use othe summatio fomulae (icludig special cases of the 6ψ 6 summatio fomula ad Jackso s 6φ 5 summatio fomula) to deive some of ou ifiite poducts We use the ew Bailey pais ad/o the summatio methods metioed above to give ew poofs of some geeal seies-poduct idetities due to Ramauja Adews ad othes We also deive a ew geeal seies-poduct idetity oe which may be egaded as a pate to oe of the Ramauja idetities We also fid ew tasfomatio fomulae betwee basic hypegeometic seies ew idetities of Roges-Ramauja type ad ew false theta seies idetities Some of these latte ae a kid of hybid i that oe side of the idetity cosists a basic hypegeometic seies while the othe side is fomed fom a theta poduct multiplied by a false theta seies This type of idetity appeas to be ew 1 Itoductio Bailey s 6ψ 6 idetity [5] q a q a b c d e 6ψ 6 q qa a a qa/b qa/c qa/d qa/e bcde (aq aq/bc aq/bd aq/be aq/cd aq/ce aq/de q q/a q) (aq/b aq/c aq/d aq/e q/b q/c q/d q/e qa /bcde q) (11) is pobably the most impotat summatio fomula fo bilateal basic hypegeometic seies ad has may applicatios to umbe theoy ad patitios - see Adews pape [] fo some examples This summatio fomula was also the mai tool used by Slate [16 17] to deive Bailey pais ad fom these he list of 130 idetities of the Roges-Ramauja type Bailey s fomula was exteded by Shukla [14] ad Shukla s fomula was late futhe exteded by Chu [6] Let K : K(a b c d e u v q) uv (σ 3 aσ 1 ) (qσ 3 aσ 1 ) a + ()uv a σ 4 a σ a + σ 4 a + (u + v)(a + uv) (σ 3 aσ 1 ) a qσ 4 a + u + a v + a a σ 4 a qσ 4 { a bcde a bcdeq (1 u)(a u)(1 v)(a v)} 1 (1) whee fo 1 k 4 σ k deotes the k-th elemetay symmetic fuctio i {b c d e} The 1

2 Popositio 11 (Chu [6]) Fo complex umbes a b c d ad e satisfyig a /bcdeq < 1 thee holds the idetity 10ψ 10 q a q a b c d e qu qa/u qv qa/v a q a a qa/b qa/c qa/d qa/e u a/u v a/v qbcde (1 aq )(b c d e uq vq aq/u aq/v q) a (1 a)(aq/b aq/c aq/d aq/e u v a/u a/v q) qbcde + 1 ( /a)(b/a c/a d/a e/a uq/a vq/a q/u q/v q) a (1 1/a)(q/b q/c q/d q/e 1/u 1/v u/a v/a q) qbcde K (aq aq/bc aq/bd aq/be aq/cd aq/ce aq/de q q/a q) (aq/b aq/c aq/d aq/e q/b q/c q/d q/e qa /bcde q) (13) Upo lettig u v we obtai Bailey s fomula (11) while lettig u ecoves Shukla s [14] 8ψ 8 geealizatio of Bailey s fomula Fo most values of the paametes the expessio fo K is quite complicated but we ote i passig that the case b a/c esults i cosideable simplificatio Coollay 1 Fo complex umbes a c d ad e satisfyig a/deq < 1 thee holds the idetity q a q a a/c c d e qu qa/u qv qa/v a 10ψ 10 q a a cq qa/c qa/d qa/e u a/u v a/v qde (1 u/c)(1 cu/a)(1 v/c)(1 cv/a) (1 u/a)(1 u)(1 v/a)(1 v) (aq q cq/d cq/e aq/cd aq/ce aq/de q q/a q) (14) (cq aq/c aq/d aq/e cq/a q/c q/d q/e qa/de q) Sice (11) was used by Slate to deive he Bailey pais it is atual to ask if Chu s geealizatio of Bailey s fomula at (13) ca be used similaly to poduce ay ew iteestig esults The peset pape is i pat a ivestigatio of that questio Oe obsevatio we make is that most of Slate s Bailey pais ae special cases of moe geeal Bailey pais cotaiig oe o moe fee paametes - see Coollaies 8 ad 13 Slate could have deived these moe geeal pais heself but it would seem that she was pimaily iteested i the special cases which would lead to idetities of the Roges-Ramauja type We also ote that Slate used the Jacobi Tiple Poduct idetity to deive he ifiite poducts I the peset pape we use othe summatio fomulae icludig special cases of the 6 ψ 6 - ad 6 φ 5 summatio fomulae to deive some ifiite poducts I the tasfomatio at (5) below Slate essetially employed thee cases (y z ad y ± aq z ) to make the seies cotaiig the α sequeces summable I the peset pape we also exploe additioal cases i the pocess discoveig some iteestig ew idetities (see Sectio 3 below) Ou esults iclude ew poofs of some geeal seies-poduct idetities due to Ramauja Adews ad othes ad also a ew geeal seies-poduct idetity (q/z q) +1 (z q) q + (q q) +1 (zq q/z q 3 q 3 ) (q q) (15) which may be egaded as a pate to a idetity equivalet to oe ecoded by Ramauja i his lost otebook [4 p 99 Ety 531 with a z/q ad q q thoughout]: (q/z q) (z q) q (q q) (zq q /z q 3 q 3 ) (q q) (16) We also fid ew tasfomatio fomulae betwee basic hypegeometic seies ew idetities of Roges-Ramauja type ad ew false theta seies idetities Some of these latte ae a kid of hybid i that oe side of the idetity cosists a basic hypegeometic seies while the othe side is fomed fom a theta poduct multiplied by a false theta seies Fo example P q + q6 + ( 8+4 ) (17) ( +1 )(q q) (q q) This type of idetity appeas to be ew

3 We employ the usual otatios: (a q) : (1 a)(1 aq) (1 aq 1 ) (a 1 a a j q) : (a 1 q) (a q) (a j q) (a q) : (1 a)(1 aq)(1 aq ) ad (a 1 a a j q) : (a 1 q) (a q) (a j q) Fo late use we ecall that a pai of sequeces (α β ) that satisfy α 0 1 ad is temed a Bailey pai elative to a β j0 α j (q q) j(aq q) +j (18) Bailey Pais fom Chu s Extesio of the 6 ψ 6 summatio fomula 1 Geeal Bailey pais We fist coside the case e a ad d q N i Chu s fomula These same substitutios wee made by Slate i (11) ad gave ise to a quite geeal Bailey pai (see below) which i tu led to may ew idetities of Roges-Ramauja type Let K be defied as at (1) Theoem 1 The pai of sequeces (α β ) is a Bailey pai with espect to a whee α : (q a q a a b c qu qa/u qv qa/v q) a ( a q ( 3)/ (1) a qa/b qa/c u a/u v a/v q q) bc (aq/bc q) β : K 1 () (aq/b aq/c q q) whee K 1 () : K(a b c q a u v q) Poof Fist set e a so that all the tems with egative idex i the bilateal sum at (13) become zeo Next fo each o-egative itege set d q so that the sum o the left side of (13) becomes a fiite sum with the summatio idex uig fom 0 to Simplify the esultig poduct o the ight side of (13) ad use the idetity (see [7 (I10) page 351]) (q q) j (q q) q j(j 1)/ (q q) j( q ) j () to modify the seies side The esult the follows fom (18) afte some simple maipulatios otig also that K 1(0) 1 so that the equiemet α 0 β 0 1 is satisfied The expessio fo K 1 () above is geeally quite complicated ad thus so also is the expessio fo the β Howeve as was also the case fo K above if we set b a/c the K 1 () simplifies cosideably Oe ca check pefeably usig a compute algeba system that K(a a/c c q a u v q) >< (c u)(c v)(a cu)(a cv) c (a u)(a v)(1 u)(1 v) (1 c)(c a)(c aq)(1 cq)uv c q(a u)(1 u)(a v)(1 v) 1 (3) >: (c u)(c v)(a cu)(a cv) c (a u)(a v)(1 u)(1 v) > 1 Upo lettig u v i the Bailey pai i Theoem 1 we ecove the followig Bailey pai (with espect to a) due to Slate α (1 aq )(a b c q) (1 a)(aq/b aq/c q q) a q ( +)/ (4) bc 3

4 (aq/bc q) β (aq/b aq/c q q) which is implicitly cotaied i Equatio (41) of [16] It would appea that Slate s picipal motivatio was to pove idetities of the Roges-Ramauja type so this Bailey pai ad othe geeal pais metioed below wee ot stated explicitly by he i [16] ad [17] whee she istead listed may special cases of them Howeve they could all have bee easily deived by he usig the same methods she used to deive the special cases We emak i passig that all the Bailey pais i Slate s B F ad H tables as well as pais E(3) E(6) ad E(7) (see [16 page 468]) ae deived fom the Bailey pai at (4) Slate also showed [16 Equatio (13) o page 46] that if (α β ) is a Bailey pai with espect to a the fo o-zeo complex umbes y ad z aq (y z q) β yz (aq/y aq/z q) (aq aq/yz q) (y z q) aq α (5) (aq/y aq/z q) yz A fiite geealizatio of this idetity is of couse implied by Bailey s Lemma (see fo example Theoem 13 i [3]) amely if (α β ) is a Bailey pai with espect to a ad N is a o-egative itege the N N q ( 3)/ α (6) (y z q N q) (yzq N /a q) q β (aq/y aq/z q) N (aq aq/yz q) N (y z q N q) aq N (aq/y aq/z aq 1+N q) yz P P The idetity at (5) is a paticula case of the Bailey Tasfom: if β δ u v + the α γ β δ 0 αu v+ ad γ I the peset pape fo ease of otatio we will efe to (5) as the Bailey Tasfom If we set b a/c i the Bailey pai fom Theoem 1 ad the substitute this pai ito (5) ad (6) we get the followig uusual basic hypegeometic idetities Coollay Let N 1 be a itege The N (c u)(c v)(a cu)(a cv) (y z q N q) 1 + c (a u)(a v)(1 u)(1 v) 1 cq aq q a(c a)(1 c)(1 y)(1 z) N uv c yzq N c(a u)(a v)(1 u)(1 v) (yz aq N ) q a aq y aq z N q N q a q a a a + c c qu qa u qv qa v y z q N q aq aq a aq N yz N q qa a cq c u a u v a v aq y aq (7) z yz aq1+n q q (c u)(c v)(a cu)(a cv) 1 + c (a u)(a v)(1 u)(1 v) 1 aq y aq z q + aq aq yz q (y z q) cq aq c q aq yz a(c a)(1 c)(1 y)(1 z)uv c(a u)(a v)(1 u)(1 v)yz q a q a a a c c qu qa u qv qa v y z q a qa a cq c u a u v a v aq y aq z q q a q ( 1)/ (8) yz Note that settig c a i (7) gives the q-pfaff-saalschütz sum (see [7 page 355 II1]) while settig c u gives the idetity (fo N 0) N q a q a a qv qa v y z q N q a a a v v aq y aq z aq1+n q q aq N yz " 1 + a(1 y)(1 z) N v (a v)(1 v) (yz aq N ) # aq aq yz N q aq y aq (9) z N q 4

5 Mod 3 Bailey pais We cotiue to follow i Slate s footsteps this time makig the same substitutios i Chu s idetity (13) that she did i (11) to poduce the Bailey pais i he A table Let K be as defied at (1) Theoem 3 (i) Let K 0() : K(a q q 1 q a u v q 3 ) The the pai of sequeces (α β ) is a Bailey pai with espect to a a whee α 0 β 0 1 α 3±1 0 ad α 3 1 aq6 1 a a q 3 u q 3 v aq 3 /u aq 3 /v q 3 u v a/u a/v q 3 q 3 q 9 15 ( a) (10) (a q 3 ) β K 0 () (a q) (q q) Let K 0() : K(aq q q 1 q aq u v q 3 ) The (ii) the pai of sequeces (α β ) is a Bailey pai with espect to a a whee α 0 β 0 1 α ad α 3 aq q3 u q 3 v aq 4 /u aq 4 /v q 3 u v aq/u aq/v q 3 q 3 q 9 13 ( a) (11) α 3+1 aq q3 u q 3 v aq 4 /u aq 4 /v q 3 u v aq/u aq/v q 3 q 3 β K 0() (aq q 3 ) (aq q) (q q) q 9 +1 ( a) +1 (iii) the pai of sequeces (α β ) is a Bailey pai with espect to a a whee α 0 β 0 1 α ad α 3 aq q3 u q 3 v aq 4 /u aq 4 /v q 3 u v aq/u aq/v q 3 q 3 q 9 7 ( a) (1) α 3+1 aq q3 u q 3 v aq 4 /u aq 4 /v q 3 u v aq/u aq/v q 3 q 3 q 9 7 ( a) β K 0() (aq q 3 ) q (aq q) (q q) Poof Set e a i (13) so that all the tems of egative idex vaish The eplace q with q 3 set b q c q 1 ad d q The afte some simple maipulatios (13) becomes /3 a q 3 u q 3 v aq 3 /u aq 3 /v q aq 6 1 a (q q) 3 (aq +1 q) 3 u v a/u a/v q 3 q 3 aq 3 6 K0() (a q3 ) (aq q) (a q) Apply () to the (q q) 3 facto divide both sides by (aq q) (q q) to get /3 1 aq 6 ( 1) q (9 15)/ a q 3 u q 3 v aq 3 /u aq 3 /v q 3 1 a (aq q) 0 +3(q q) 3 u v a/u a/v q 3 q 3 a (a q 3 ) K 0() (13) (a q) (q q) ad (10) follows Fo the othe two pais eplace a with aq i (13) ad (11) follows fom the idetity (1 aq 6+1 )q (9 13)/ (aq q) +3+1 (q q) 3 while (1) follows fom the idetity (1 aq 6+1 )q (9 13)/ (aq q) +3+1 (q q) 3 q (9 13)/ (aq q) +3 (q q) 3 q q (9 7)/ (aq q) +3 (q q) 3 a q (9 )/+1 (aq q) +3+1 (q q) 3 1 q q (9 7)/ (aq q) +3+1 (q q) 3 1 Remak 4 K 0 (0) K 0 (1) K 0 () 1 5

6 Theoem 5 Let K () : K(q q q 1 q e u v q 3 ) ad suppose e 1/q The (i) the pai of sequeces (α β ) is a Bailey pai with espect to a 1 whee α 0 β 0 1 ad α 3 ( 1) q 9 11 e q 3 u q 3 v q 4 /u q 4 /v q 3 q 4 /e u v q/u q/v q 3 + q 9 5 e α 3+1 ( 1) +1 q e q3 u q 3 v q 4 /u q 4 /v q 3 q 4 /e u v q/u q/v q 3 e α 3 1 ( 1) +1 q e/q q u q v q 3 /u q 3 /v q 3 q 3 /e u/q v/q 1/u 1/v q 3 e (q /e q 3 ) β K () (q q) (q /e q) e/q q u q v q 3 /u q 3 /v q 3 q 3 /e u/q v/q 1/u 1/v q 3 (14) e (ii) the pai of sequeces (α β ) is a Bailey pai with espect to a 1 whee α 0 β 0 1 ad α 3 ( 1) q 9 5 α 3+1 ( 1) +1 q 9 5 α 3 1 ( 1) +1 q 9 11 (q /e q 3 ) q β K () (q q) (q /e q) e q 3 u q 3 v q 4 /u q 4 /v q 3 q 4 /e u v q/u q/v q 3 + q 9 11 e e q 3 u q 3 v q 4 /u q 4 /v q 3 q 4 /e u v q/u q/v q 3 e e/q q u q v q 3 /u q 3 /v q 3 q 3 /e u/q v/q 1/u 1/v q 3 e e/q q u q v q 3 /u q 3 /v q 3 q 3 /e u/q v/q 1/u 1/v q 3 (15) e (iii) the pai of sequeces (α β ) is a Bailey pai with espect to a q whee α 0 β 0 1 ad α 3 ( 1) 6+1 q 9 11 α α 3 1 ( 1) (q /e q 3 ) β K () (q q) (q /e q) e q 3 u q 3 v q 4 /u q 4 /v q 3 q 4 /e u v q/u q/v q 3 (16) e q e/q q u q v q 3 /u q 3 /v q 3 q 3 /e u/q v/q 1/u 1/v q 3 e Remak 6 The coditio e 1/q is ecessay to esue that β 0 1 Poof Replace q with q 3 i (13) ad the set a q b q c q 1 ad d q Oe easily checks that the ight side simplifies to give K (q () q q) (q /e q 3 ) (q q) (q /e q) O the seies side the choices fo the paametes foce the seies to temiate above ad below ad we get afte some elemetay maipulatios that the seies becomes /3 6+1 (q q) 3 e q 3 u q 3 v q 4 /u q 4 /v q 3 q 3 4 (q + q) 3 q 4 /e u v q/u q/v q 3 e +1 3 (+1)/3 Next we apply () to the (q q) 3 facto above ad eaage tems to get /3 ( 6+1 )( 1) q (9 11)/ e q 3 u q 3 v q 4 /u q 4 /v q 3 (q q) +3+1 (q q) 3 e Notig that fo abitay o-zeo y ad z (y q) (z q) q 4 /e u v q/u q/v q 3 K()(q /e q 3 ) (q q) (q /e q) (q/z q) z (q/y q) y (17) 6

7 we get that (q /e q 3 ) K () (q q) (q /e q) /3 0 /3 0 + ( 6+1 )( 1) q (9 11)/ (q q) +3+1 (q q) 3 e ( 6+1 )( 1) q (9 5)/ (q q) 3+1(q q) +3e q (9 11)/ (q q) +3(q q) 3 (+1) 3 1 q (9 5)/ (q q) 3 (q q) +3 whee the last equality follows fom the idetities e q 3 u q 3 v q 4 /u q 4 /v q 3 q 4 /e u v q/u q/v q 3 e/q q u q v q 3 /u q 3 /v q 3! q (9 +)/+1 (q q) +3+1(q q) 3 1 q 3 /e u/q v/q 1/u 1/v q 3! q (9 17)/+1 (q q) 3+1 (q q) +3 1 ( 1) e q3 u q 3 v q 4 /u q 4 /v q 3 q 4 /e u v q/u q/v q 3 ( 1) e/q q u q v q 3 /u q 3 /v q 3 q 3 /e u/q v/q 1/u 1/v q 3 ( 6+1 )q 9 11 q 9 11 ( +3+1 ) q ( 3 ) (18) ( 6+1 )q 9 5 q 9 5 ( 3+1 ) q ( +3 ) That (14) gives a Bailey pai ow follows fom the defiitio of a Bailey pai at (18) also otig that K (0) K (1) 1 If istead of usig the idetities at (18) we use the idetities ( 6+1 )q 9 11 q 9 5 (( +3+1 ) ( 3 )) (19) ( 6+1 )q 9 5 q 9 11 (( 3+1 ) ( +3 )) we get that the pai at (15) is a Bailey pai The poof of (16) follows fom the ight side of the fist equality followig (17) upo settig (q q) +3+1 (q q) 3 ()(q q) +3 (q q) 3 i the fist sum ad (q q) 3+1 (q q) +3 ()(q q) 3+1 (q q) +3 1 i the secod sum If we begi by settig a q istead of a q but keep the same choices b q c q 1 ad d q i (13) the we get Theoems 7 followig Theoem 7 Let K 3 () : K(q q q 1 q e u v q 3 ) The (i) the pai of sequeces (α β ) is a Bailey pai with espect to a q whee α 0 β 0 1 ad α 3 ( 1) q 9 7 e q 3 u q 3 v q 5 /u q 5 /v q 3 q 5 /e u v q /u q /v q 3 (0) e α 3+1 ( 1) +1 q e q 3 u q 3 v q 5 /u q 5 /v q 3 q 5 /e u v q /u q /v q 3 + ( 1) q e α 3 1 ( 1) q 9 7 (q 4 /e q 3 ) β K 3() (q q) (q 3 /e q) e/q qu qv q 3 /u q 3 /v q 3 q 3 /e u/q v/q 1/u 1/v q 3 e 9 3 e/q qu qv q 3 /u q 3 /v q 3 +1 q 3 /e u/q v/q 1/u 1/v q 3 +1 e+1 (ii) the pai of sequeces (α β ) is a Bailey pai with espect to a q whee α 0 β 0 1 ad α 3 ( 1) q 9 α 3+1 ( 1) +1 q 9 e q 3 u q 3 v q 5 /u q 5 /v q 3 q 5 /e u v q /u q /v q 3 (1) e e q 3 u q 3 v q 5 /u q 5 /v q 3 q 5 /e u v q /u q /v q 3 + ( 1) q e 9 +5 e/q qu qv q 3 /u q 3 /v q 3 +1 q 3 /e u/q v/q 1/u 1/v q 3 +1 e+1 7

8 α 3 1 ( 1) q 9 13 (q 4 /e q 3 ) q β K 3() (q q) (q 3 /e q) e/q qu qv q 3 /u q 3 /v q 3 q 3 /e u/q v/q 1/u 1/v q 3 (iii) the pai of sequeces (α β ) is a Bailey pai with espect to a q whee α 0 β 0 1 ad α 3 ( 1) 6+ q 9 7 α 3+1 ( 1) 6+4 q 9 α (q 4 /e q 3 ) β K 3 () (q q) (q 3 /e q) e e q 3 u q 3 v q 5 /u q 5 /v q 3 q 5 /e u v q /u q /v q 3 () e e/q qu qv q 3 /u q 3 /v q 3 +1 q 3 /e u/q v/q 1/u 1/v q 3 +1 e+1 Poof The poof paallels that of Theoem 5 except that afte aivig at the idetity ( 6+ )( 1) q (9 7)/ (q q) +3+1 (q q) 3 e e q 3 u q 3 v q 5 /u q 5 /v q 3 q 5 /e u v q /u q /v q 3 K3()(q4 /e q 3 ) (q q) (q 3 /e q) (3) ad the sepaatig the sum ito two sums ( 0 ad < 0) as peviously we istead employ the idetities ( 6+ )q 9 7 q 9 7 ( +3+ ) q ( 3 ) (4) ( 6+ )q 9 7 q 9 7 ( 3+ ) q ( +3 ) to get (0) The esult follows as i Theoem 5 except that it is ecessay to e-idex oe of the fou sums (by eplacig with + 1) Fo (1) we istead use the idetities ( 6+ )q 9 7 q 9 (( +3+ ) ( 3 )) (5) ( 6+ )q 9 7 q 9 13 (( 3+ ) ( +3 )) The poof of () is like the poof of (16) i Theoem 5 except that afte sepaatig the sum at (3) ito two sums accodig to 0 o < 0 ad the eplacig with fo the sum with < 0 we use the idetities (q q) +3+1 (q q) 3 ( )(q 3 q) +3 (q q) 3 ad (q q) 3+1 (q q) 3 ( )(q q) 3+ (q 3 q) +3 ad fially e-idex i the latte case by eplacig with + 1 The ie Bailey pais i the ext coollay deive espectively fom the pais i Theoems 3-7 by lettig u v i each case Coollay 8 The sequeces (α β ) below ae Bailey pais with espect to the stated values of a whee α 3 1 aq6 1 a α 3±1 0 β (a q 3 ) q 9 3 (q 3 q 3 ( a) (6) ) (a q 3 ) (a q) (q q) with espect to a a α 3 aq q3 q 3 q 3 q 9 ( a) (7) α 3+1 aq q3 q 3 q 3 α q ( a) +1 8

9 (aq q 3 ) β with espect to a a (aq q) (q q) α 3 aq q3 q 3 q 3 q 9 +5 ( a) (8) α 3+1 aq q3 q 3 q 3 q 9 +5 ( a) α β α 3 ( 1) (aq q3 ) q (aq q) (q q) with espect to a a q 9 /+/ (e q 3 ) /+7/! + q9 (e/q q 3 ) (9) (q 4 /e q 3 ) e (q 3 /e q 3 ) e α 3+1 ( 1)+1 q 9 /+13/+1 (e q 3 ) (q 4 /e q 3 ) e α 3 1 ( 1)+1 q 9 / 5/+1 (e/q q 3 ) (q 3 /e q 3 ) e β α 3 ( 1) (q /e q 3 ) (q q) (q /e q) with espect to a 1 q 9 /+7/ (e q 3 ) /+/! + q9 (e/q q 3 ) (30) (q 4 /e q 3 ) e (q 3 /e q 3 ) e α 3+1 ( 1)+1 q 9 /+7/ (e q 3 ) (q 4 /e q 3 ) e α 3 1 ( 1)+1 q 9 /+/ (e/q q 3 ) (q 3 /e q 3 ) e β q (q /e q 3 ) (q q) (q /e q) with espect to a 1 α 3 ( 1) 6+1 q 9 + α α 3 1 ( 1) β e q 3 q 4 /e q 3 (31) e q e/q q 3 q 3 /e q 3 e (q /e q 3 ) (q q) (q /e q) with espect to a q α 3 ( 1) q 9 /+5/ (e q 3 ) (3) (q 5 /e q 3 ) e α 3+1 ( 1) q 9 /+11/+3 (e/q q 3 ) +1 + ( 1)+1 q 9 /+17/+ (e q 3 ) (q 3 /e q 3 ) +1 e +1 (q 5 /e q 3 ) e α 3 1 ( 1) q 9 /+5/ (e/q q 3 ) (q 3 /e q 3 ) e β (q 4 /e q 3 ) (q q) (q 3 /e q) with espect to a q 9

10 α 3 ( 1) q 9 /+11/ (e q 3 ) (33) (q 5 /e q 3 ) e α 3+1 ( 1) q 9 /+17/+4 (e/q q 3 ) +1 + ( 1)+1 q 9 /+11/ (e q 3 ) (q 3 /e q 3 ) +1e +1 (q 5 /e q 3 ) e α 3 1 ( 1) q 9 / / (e/q q 3 ) (q 3 /e q 3 ) e β q (q 4 /e q 3 ) (q q) (q 3 /e q) with espect to a q α 3 ( 1) 6+ q 9 +5 α 3+1 ( 1) 6+4 q 9 α β e q 3 q 5 /e q 3 (34) e e/q q 3 +1 q 3 /e q 3 +1 e+1 (q 4 /e q 3 ) (q q) (q 3 /e q) with espect to a q All eight pais i Slate s A table ([16 page 463]) ad all six o he J list ([17 pp ]) ae deived fom the five Bailey pais (6) (9) (30) (3) (33) above fo paticula values of e Slate did ot wite dow these geeal pais above explicitly but she could easily have deived them by the same methods she used to deive the special cases Noe of Slate s pais aise as special cases of (7) (8) (31) o (34) although the special case e q of (31) was give i [9] Howeve specializig the paametes give Bailey pais which the give ise to some of the seies-poduct idetities o Slate s list showig that diffeet Bailey pais may lead the same idetity of Roges-Ramauja type Each of the ie Bailey pais above also gives ise to a tasfomatio betwee basic hypegeometic seies upo substitutig the pai ito (5) The pai at (6) fo example gives the followig idetity Coollay 9 (y z q) (a q 3 ) (a q) (q q) 3 Mod Bailey Pais aq (aq/y aq/z q) yz (aq aq/yz q) (1 aq 6 )(y z q) 3 (a q 3 ) a 4 q 9 /+3/ 1 3 (35) (1 a)(aq/y aq/z q) 3(q 3 q 3 ) yz We ext coside how Slate poduced the Bailey pais i the G- C- ad I tables of [16 17] Theoem 10 Let K 4 () : K(a q q 1 d a u v q ) The (i) the pai of sequeces (α β ) is a Bailey pai with espect to a a whee α 0 β 0 1 ad α 1 aq4 1 a α 1 0 a d q u q v aq /u aq /v q q 4 a aq /d u v a/u a/v q q d (36) (aq/d q ) β K 4() (aq q ) (aq/d q q) Let K 4() : K(aq q q 1 d aq u v q ) The (ii) the pai of sequeces (α β ) is a Bailey pai with espect to a a whee α 0 β 0 1 ad α aq d q u q v aq 3 /u aq 3 /v q 3 q a aq 3 /d u v aq/u aq/v q q (37) d α +1 aq d q u q v aq 3 /u aq 3 /v q ++1 q a +1 aq 3 /d u v aq/u aq/v q q β K 4 () (aq /d q ) (aq q ) (aq /d q q) d 10

11 (iii) the pai of sequeces (α β ) is a Bailey pai with espect to a a whee α 0 β 0 1 ad α aq d q u q v aq 3 /u aq 3 /v q q a aq 3 /d u v aq/u aq/v q q (38) d α +1 aq d q u q v aq 3 /u aq 3 /v q q a aq 3 /d u v aq/u aq/v q q β K 4 () (aq /d q ) q (aq q ) (aq /d q q) d Poof The poof is quite simila to the poof of Theoem 3 As i the poof of that theoem set e a i (13) so that all the tems of egative idex vaish The eplace q with q set b q ad d q 1 Afte some simple maipulatios (13) becomes / a d q u q v aq /u aq /v q aq aq 4 1 a (q q) (aq +1 q) aq /du v a/u a/v q q d K 4() (aq/d q ) (aq q) (aq q ) (aq/d q) Apply () to the (q q) facto divide both sides by (aq q) (q q) to get / 1 aq 4 q 4 a d q u q v aq /u aq /v q a (aq/d q ) K4() 1 a (aq q) 0 +(q q) aq /du v a/u a/v q q (39) d (aq q ) (aq/d q q) ad the poof fo the pai at (36) follows Fo (37) ad (38) eplace a with aq i (39) ad use espectively the idetities (1 aq 4+1 )q 3 q 3 (1 aq ++1 ) aq ++1 ( ) (1 aq 4+1 )q 3 q q ((1 aq ++1 ) ( )) Theoem 11 Let K 5() : K(q q q 1 d e u v q ) The (i) the pai of sequeces (α β ) is a Bailey pai with espect to a q whee α 0 β 0 1 ad α 4+1 α d e q u q v q 3 /u q 3 /v q q q 3 /d q 3 /e u v q/u q/v q (40) (d e) d/q e/q qu qv q /u q /v q 4+1 q q /d q /e u/q v/q 1/u 1/v q (de) (q 3 /de q ) β K 5 () (q q ) (q /d q /e q) (ii) The pai of sequeces (α β ) is a Bailey pai with espect to a 1 whee α 0 β 0 1 ad α d e q u q v q 3 /u q 3 /v q q d/q e/q qu qv q /u q /v q q 3 /d q 3 /e u v q/u q/v q (d + q e) q /d q /e u/q v/q 1/u 1/v q (41) (de) α 1 d e q u q v q 3 /u q 3 /v q +1 1 q q 3 /d q 3 /e u v q/u q/v q (d d/q e/q qu qv q /u q /v q 4+1 q e) 1 q /d q /e u/q v/q 1/u 1/v q 1 (de) (q 3 /de q ) β K 5 () (q q ) (q /d q /e q) (iii) The pai of sequeces (α β ) is a Bailey pai with espect to a 1 whee α 0 β 0 1 ad α d e q u q v q 3 /u q 3 /v q q d/q e/q qu qv q /u q /v q q 3 /d q 3 /e u v q/u q/v q + q (d e) q /d q /e u/q v/q 1/u 1/v q (4) (de) α 1 d e q u q v q 3 /u q 3 /v q 4+ 1 q q 3 /d q 3 /e u v q/u q/v q (d d/q e/q qu qv q /u q /v q q e) 1 q /d q /e u/q v/q 1/u 1/v q 1 (de) β K 5 () q (q 3 /de q ) (q q ) (q /d q /e q) 11

12 Poof The poof is vey simila to the poof of Theoem 5 This time eplace q with q i (13) ad the set a q b q c q 1 The ight side simplifies to give K 5 () (q q q) (q 3 /de q ) (q q ) (q /d q /e q) Afte some elemetay maipulatios the seies becomes / (+1)/ 4+1 (q q) d e q u q v q 3 /u q 3 /v q (q + q) q 3 /d q 3 /e u v q/u q/v q We apply () to the (q q) facto above ad eaage tems to get / +1 ( 4+1 )q d e q u q v q 3 /u q 3 /v q ()(q q) + (q q) d e q 3 /d q 3 /e u v q/u q/v q Afte applyig (17) to the tems of egative idex i the sum above we get that (q 3 /de q ) K 5 () (q q ) (q /d q /e q) / 0 q 1 de ( 4+1 )q d e q u q v q 3 /u q 3 /v q ()(q q) +(q q) d e q 3 /d q 3 /e u v q/u q/v q +1 1 The pai at (40) ow follows afte some simple maipulatios otig that K 5()(q 3 /de q 3 ) (q q ) (q /d q /e q) ( 4 1 )q 4+1 d/q e/q qu qv q /u q /v q ()(q q) (q q) + d e q /d q /e u/q v/q 1/u 1/v q (43) (q q) (q q) + (q q) +1 (q q) + 1 The pai at (41) follows fom (43) upo absobig the facto i the deomiatos thee employig the idetities ( 4+1 )q q ( ++1 ) q ++1 ( ) (44) ( 4 1 )q 4+1 q 4+1 ( + ) q ( +1 ) i the way simila to the way that the pai of idetities at (18) was used i the poof of Theoem 5 ad fially e-idexig oe of the esultig sums (by eplacig with 1) The poof fo the pai at (4) is simila to the poof fo the pai at (41) except we employ the idetities ( 4+1 )q q (( ++1 ) ( )) (45) ( 4 1 )q 4+1 q (( + ) ( +1 )) As with K 0 - K 3 K 4 ad K 5 ae i geeal quite complicated but simplify cosideably fo paticula values of the paametes leadig (as was the case i Coollay ) to tasfomatios of basic hypegeometic seies We give oe example Coollay 1 If a y z q C such that q < 1 ad oe of the deomiatos below vaish the (y z q) ( 1/q q ) aq (aq/y aq/z q) 1 aq 4 (q q ) (aq q ) yz (aq aq/yz q) 1 a (y z q) (a aq 4 q 4 ) a q (aq/y aq/z q) ( a q 4 q 4 ) y z Poof I the Bailey pai at (36) set d a/q u q ad v i a so that K 4 () takes the value K a q q 1 a q a q i a q q 1 (q + 1) q q (q + 1) (q 1 + 1) (q ) (46) Substitute the esultig Bailey pai ito (5) ad the esult follows afte some elemetay q-poduct maipulatios 1

13 The Bailey pais i Slate s G- C- ad I tables as well as pais E(1) E() E(4) ad E(5) (see [16 pages 469 ad 470]) ae deived fom the ext six Bailey pais These i tu ae deived fom the pais i Theoems 10 ad 11 by lettig u v Coollay 13 The sequeces (α β ) below ae Bailey pais with espect to the stated values of a whee α 0 β 0 1 ad α 1 aq4 a d q a q 1 a aq /d q q (47) d α 1 0 β (aq/d q ) (aq q ) (aq/d q q) with espect to a a α aq d q + q a aq 3 /d q q (48) d α +1 aq d q +5+1 q a +1 aq 3 /d q q β d (aq /d q ) (aq q ) (aq /d q q) with espect to a a α aq d q +3 q a aq 3 /d q q (49) d α +1 aq d q +3 q a aq 3 /d q q β d (aq /d q ) q (aq q ) (aq /d q q) with espect to a a α (4+1 )q + (d q ) (e q ) ()(q 3 /d q ) (q 3 /e q ) d e (50) α 1 (4 1 )q +1 (d/q q ) (e/q q ) ()(q /d q ) (q /e q ) d e β (q 3 /de q ) (q q ) (q /d q) (q /e q) with espect to a q +4 α q (d/q q ) (e/q q ) + (q /d q ) (q /e q ) d e + q (d q ) (e q ) (q 3 /d q ) (q 3 /e q ) d e (51) +4+3 α +1 q (d/q q ) +1 (e/q q ) +1 (q /d q ) +1 (q /e q ) +1 d +1 e β +1 q (q 3 /de q ) (q q ) (q /d q) (q /e q) with espect to a (d q ) (e q ) (q 3 /d q ) (q 3 /e q ) d e + α q (d/q q ) (e/q q ) +4 (q /d q ) (q /e q ) d e + q (d q ) (e q ) (q 3 /d q ) (q 3 /e q ) d e (5) +6+4 α +1 q (d/q q ) +1 (e/q q ) +1 (q /d q ) +1 (q /e q ) +1 d +1 e β +1 q q (q 3 /de q ) (q q ) (q /d q) (q /e q) with espect to a 1 +4 (d q ) (e q ) (q 3 /d q ) (q 3 /e q ) d e 13

14 4 Mod 4 Bailey Pais We cotiue to follow i Slate s path [16] ext cosideig how she poduced the Bailey pais i he K table Theoem 14 Let K 6() : K(q q q 1 q q 3 u v q 4 ) fo 3 ad K 6(0) K 6(1) K 6() 1 The (i) the pai of sequeces (α β ) is a Bailey pai with espect to a 1 whee α 0 β 0 1 α 4+ 0 ad α 4 q4 u q 4 v q 5 /u q 5 /v q 4 u v q/u q/v q 4 q q3 u q 3 v q 4 /u q 4 /v q 4 u/q v/q 1/u 1/v q 4 q 8 6 (53) α 4+1 q4 u q 4 v q 5 /u q 5 /v q 4 u v q/u q/v q 4 q 8 +1 α 4 1 q3 u q 3 v q 4 /u q 4 /v q 4 u/q v/q 1/u 1/v q 4 q β K 6 () ( q q ) 1 (q q) (ii) the pai of sequeces (α β ) is a Bailey pai with espect to a 1 whee α 0 β 0 1 α 4+ 0 ad α 4 q4 u q 4 v q 5 /u q 5 /v q 4 u v q/u q/v q 4 q q3 u q 3 v q 4 /u q 4 /v q 4 u/q v/q 1/u 1/v q 4 q 8 10 (54) α 4+1 q4 u q 4 v q 5 /u q 5 /v q 4 u v q/u q/v q 4 q 8 6 α 4 1 q3 u q 3 v q 4 /u q 4 /v q 4 u/q v/q 1/u 1/v q 4 q 8 10 β K 6() q ( q q ) 1 (q q) Poof The poof is iitially simila to the poof of Theoem 3 except we eplace q with q 4 ad set b q c q 1 d q ad e q 3 Istead of (13) we aive at q 4 u q 4 v aq 4 /u aq 4 /v q 4 Z 1 aq 8 1 a a q 8 1 (aq q) +4(q q) 4 u v a/u a/v q 4 K(a q q 1 q q 3 u v q 4 ) (q4 /a a/q a aq q 4 ) (q q q 3 a /q q 4 ) ( a/q q ) (a q) (55) Next sepaate the sum ito tems of positive ad egative idex e-idex the latte sum by eplacig with (ad also usig (17)) to get q 4 u q 4 v aq 4 /u aq 4 /v q aq 8 1 a a q 8 1 (aq q) +4(q q) aq 8 1 a u v a/u a/v q 4 a q 8 4 (aq q) 4 (q q) +4 q 4 /u q 4 /v uq 4 /a vq 4 /a q 4 1/u 1/v u/a v/a q 4 K(a q q 1 q q 3 u v q 4 ) (q4 /a a/q a aq q 4 ) (q q q 3 a /q q 4 ) ( a/q q ) (a q) (56) The poof fo the Bailey pai at (53) the follows upo settig a q simplifyig the poduct side ad usig the idetities ( 8+1 )q 8 10 q 8 10 ( +4+1 ) q 8 +1 ( 4 ) ( 8+1 )q 8 6 q 8 6 ( 4+1 ) q ( +4 ) The poof fo the Bailey pai at (54) is simila except we use the idetities ( 8+1 )q 8 10 q q 8 6 (( +4+1 ) ( 4 )) ( 8+1 )q 8 6 q q 8 10 (( 4+1 ) ( +4 )) 14

15 Theoem 15 Let K 7 () : K(q q q 1 q q 3 u v q 4 ) fo 0 The (i) the pai of sequeces (α β ) is a Bailey pai with espect to a q whee α 0 β 0 1 ad α 4 q4 u q 4 v q 6 /u q 6 /v q 4 u v q /u q /v q 4 q 8 8 (57) α 4+1 q4 u q 4 v q 6 /u q 6 /v q 4 u v q /u q /v q 4 q 8 + α 4 1 q u q v q 4 /u q 4 /v q 4 u/q v/q 1/u 1/v q 4 q 8 8 α 4 q u q v q 4 /u q 4 /v q 4 u/q v/q 1/u 1/v q 4 q β K 7() ( q q ) (q q) (ii) the pai of sequeces (α β ) is a Bailey pai with espect to a q whee α 0 β 0 1 ad α 4 q4 u q 4 v q 6 /u q 6 /v q 4 u v q /u q /v q 4 q 8 4 (58) α 4+1 q4 u q 4 v q 6 /u q 6 /v q 4 u v q /u q /v q 4 q 8 4 α 4 1 q u q v q 4 /u q 4 /v q 4 u/q v/q 1/u 1/v q 4 q 8 1 α 4 q u q v q 4 /u q 4 /v q 4 u/q v/q 1/u 1/v q 4 q 8 1 β K 7 () q ( q q ) (q q) Poof Set a q i (56) simplify the poduct side ad absob the tems o the sum side ito what wee peviously the (aq q) +4 ad (aq q) 4 factos The pai at (57) follows afte usig the idetities ( 8+ )q 8 8 q 8 8 ( +4+ ) q 8 + ( 4 ) ( 8+ )q 8 8 q 8 8 ( 4+ ) q ( +4 ) The pai at (58) follows similaly afte employig the idetities ( 8+ )q 8 8 q q 8 4 (( +4+ ) ( 4 )) ( 8+ )q 8 8 q q 8 1 (( 4+ ) ( +4 )) Theoem 16 Let K 8 (0) 1 K 8 (1) (q u) q u (q v) q v (q 3 u) (u 1) (q 3 v) (v 1) (q 1)4 (q + 1) q + q + 1 uv q (q 3 u) (u 1) (q 3 v) (v 1) ad K 8 () : K(q 3 q q 1 q q 3 u v q 4 ) fo The (i) the pai of sequeces (α β ) is a Bailey pai with espect to a q whee α 0 β 0 1 ad α 4 q4 u q 4 v q 7 /u q 7 /v q 4 u v q 3 /u q 3 /v q 4 q 8 6 (59) α 4+1 q4 u q 4 v q 7 /u q 7 /v q 4 u v q 3 /u q 3 /v q 4 α q q4 /u q 4 /v qu qv q /u 1/v u/q 3 v/q 3 q 4 q

16 α 4 q4 /u q 4 /v qu qv q 4 1/u 1/v u/q 3 v/q 3 q 4 q 8 10 β K 8() ( q q ) (q 3 q) (ii) the pai of sequeces (α β ) is a Bailey pai with espect to a q whee α 0 β 0 1 ad α 4 q4 u q 4 v q 7 /u q 7 /v q 4 u v q 3 /u q 3 /v q 4 q 8 (60) α 4+1 q4 u q 4 v q 7 /u q 7 /v q 4 u v q 3 /u q 3 /v q 4 α α 4 q4 /u q 4 /v qu qv q 4 1/u 1/v u/q 3 v/q 3 q 4 q 8 14 β K 8 ()q ( q q ) (q 3 q) q 8 q4 /u q 4 /v qu qv q /u 1/v u/q 3 v/q 3 q 4 q Remak 17 The easo K 8(1) does ot fit ito the fomula fo K 8() fo > 1 is that it is ecessay to be caeful with the ode i which (afte eplacig q with q 4 ) we set a q 3 b 1/q c 1 d q ad e q i ode to get the seies at (13) to covege It is easy to see that makig those substitutios simultaeously gives a /q 4 bcde 1 cotadictig the equiemet that a /q 4 bcde < 1 i the seies Oe way aoud this is to fist set c 1 ad b a/q 4 causig the seies to temiate above ad below afte which the othe eplacemets ca be made Note also that with the stated choices fo the paametes the a bcdeq 4 facto i the deomiato of the expessio fo K at (1) vaishes ad i fact these choices also cause the umeato of K to vaish It is cacellig these zeo factos that make the expessio fo K 8 (1) diffeet The ode of substitutios descibed above causes K(q 3 q 1 1 q q u v q 4 ) to have the value assiged to K 8(1) wheeas fo > 1 K(q 3 q q 1 q q 3 u v q 4 ) is idepedet of the ode of substitutios We also ote that a simila situatio occus i some of the othe theoems above Poof The poof is simila to the poof of Theoem 15 except we set a q 3 i (56) The pai at (59) follows afte usig the idetities ( 8+3 )q 8 6 q 8 6 ( +4+3 ) q ( 4 ) ( 8+3 )q 8 10 q 8 10 ( 4+3 ) q ( +4 ) ad the eplacig with +1 i the sum coespodig to the secod tem i the secod idetity above (so that this sum implicitly defies pat of α 4+1 istead of α 4 3) The pai at (60) follows similaly afte usig the idetities ( 8+3 )q 8 6 q q 8 (( +4+3 ) ( 4 )) ( 8+3 )q 8 10 q q 8 14 (( 4+3 ) ( +4 )) The six Bailey pais i Slate s K table [16 page 471] follow fom the thee theoems i this sectio upo lettig u v Coollay 18 The followig pais of sequeces (α β ) with α 0 β 0 1 ae Bailey pais with espect to the stated value of a α 4 q 8 + q 8 + α 4+1 q α 4 1 q α 4 0 (61) 16

17 β ( q q ) 1 (q q) with espect to a 1 α 4 q q 8 α 4+1 q 8 + α 4 1 q 8 α 4 0 (6) β q ( q q ) 1 (q q) with espect to a 1 α 4 q 8 (63) α 4+1 q α 4 1 q 8 α 4 q 8 8+ β ( q q ) (q q) with espect to a q α 4 q 8 +4 α 4+1 q 8 +4 α 4 1 q 8 4 α 4 q 8 4 (64) β q ( q q ) (q q) with espect to a q α 4 q 8 + α 4+1 q q α α 4 q 8 (65) β ( q q ) (q 3 q) with espect to a q α 4 q 8 +6 α 4+1 q 8 +6 q α α 4 q 8 6 (66) β q ( q q ) (q 3 q) with espect to a q 3 Idetities of the Roges-Ramauja-Slate type ad False Theta Seies Idetities We use the Bailey pais foud i the pevious sectio to deive seveal ew seies-poduct idetities ad also to give ew poofs of some geeal idetities due to Ramauja Adews ad othes We also emak that we sometimes use a moe geeal case of the 6ψ 6 tha the Jacobi Tiple Poduct Idetity i cotast to how Slate deived he poducts 17

18 31 Geeal idetities cotaiig oe o moe fee paametes We fist give ew poofs fo two quite geeal idetities of Ramauja ([1 page 33] see also R1 ad R o page 8 of [10]) Oe easo these two idetities ae of iteest is that each leads to ifiitely may idetities of Roges- Ramauja type (set z ±q a/b whee a ad b > 0 ae iteges ad the eplace q with q b ) Theoem 31 Fo q < 1 ad z C \ {0} (q/z q) (z q) q (q q) (qz q /z q 3 q 3 ) (q q) (31) Poof Let e 0 i the Bailey pai at (31) ad the iset the esultig pai ito (5) afte settig y q/z This leads to the idetities q q (z qz qz q q)q z 6+1 q z q z 3 q (q/z z q) 3 1 q 3 (q q) qz qz q (qz q /z q) 3 1 qz qz q (q q q) 0 qz qz q 6+1 (q q q) 0 q z z q q3 3 z q4z q3 q /q 3 1 (1/z z/q q 3 ) q 3 (q z q 3 /z q 3 ) (q 4 q 3 q 3 q q 3 ) (q q q) (q 4 /z zq 3 q 3 /z q z q 3 ) The secod equality above follows afte some elemetay q-poduct maipulatios ad the last equality follows fom the fact that the two seies i the pevious equality combie to give a special case of Bailey s 6ψ 6 summatio at (11) (eplace q with q 3 let d e set a q b q/z ad c z) The esult ow follows afte some futhe elemetay maipulatios Theoem 3 Fo q < 1 ad z C \ {0} (q /z q ) (z q ) q (q q ) (q 4 q 4 ) ( q q ) (q q ) (qz q 3 /z q 4 q 4 ) (3) Poof The poof is simila to that of the theoem above Let e 0 ad set d q 3/ i the Bailey pai at (50) ad the iset the esultig pai ito (5) afte settig y q/z This leads to the idetities q q (z q) z q / (q 1/ q) (q q ) qz qz q (q q q) 0 qz qz q (q q q) 0 qz qz q q z z q q +/ qz q z q ( 1) 1 q z z q ( 1) q +/ + q z q3z q /q (q/z z q) 1q / ( 1) (qz q /z q) 1 (1/z z/q q ) ( 1) q / (qz q /z q ) (q 3 q zq 1/ q 3/ /z q q q ) (q q q) (q 3 /z zq q 3/ q /z qz q 1/ q ) The secod equality above oce agai follows afte some elemetay q-poduct maipulatios ad oce agai the two seies o the ight side i the secod equality combie to give a special case of Bailey s 6ψ 6 summatio at (11) (eplace q with q let e set a q b q/z c z ad d q 3/ ) The esult ow follows afte eplacig q with q followed by some futhe elemetay maipulatios Diffeet poofs of the esults i the two theoems above wee also give i [4] (Eties 531 ad 535) ad i [11] We ow pove a ew geeal seies-poduct idetity oe which may be egaded as a pate to Ramauja s esult i Theoem 31 18

19 Theoem 33 Fo q < 1 ad z C \ {0} (q/z q) +1(z q) q + (q q) +1 (q z q/z q 3 q 3 ) (q q) (33) Poof Let e 0 i the Bailey pai at (34) ad the iset the esultig pai ito (5) afte settig y q /z This leads to the idetities q q (z q) z q + (q q) qz q3z q q 6+ z 3 z q (q 3 q q) 0 qz q3z q 3 qz q3z q q 6+ z z q3 (q 3 q q) 0 q 3 z q5z q3 qz q3z q (q 5 q 3 q 3 q q 3 ) (q 3 q q) (q 5 /z zq 3 q 3 /z qz q 3 ) q 3 + q (q /z z q) 3+1q 3 (qz q 3 /z q) (1/z z/q q 3 ) q 3 1 1/q (qz q 3 /z q 3 ) +4+1 The secod equality above follows afte some elemetay q-poduct maipulatios ad e-idexig the secod seies (eplacig with 1) The last equality follows fom the fact that the two seies i the pevious equality combie to give aothe special case of Bailey s 6 ψ 6 summatio (11) (eplace q with q 3 let d e set a q b q /z ad c z) The esult oce agai follows afte some futhe elemetay maipulatios Remak 34 Seveal idetities o Slate s list [17] follow as special cases of (33) followig table We summaize these i the Replace q by set z to to obtai Slate s idetity q q () q q (7) (87) q iq (8) q 3 q (40) q 3 q (41) q 4 q 3 (55) (57) q e πi/3 q (9) We ext give ew poofs of Adews q-aalog of Gauss s F 1 (1/) sum ad a special case (b ) of Heie s [8] q-aalog of Gauss s sum: (a b q) c (c/a c/b q) (34) (c q q) ab (c c/ab q) We fist ecall Jackso s summatio fomula fo a vey-well-poised 6 φ 5 seies [7 p 356 Eq (II 0)] (which follows upo settig e a i (11)): Theoem 35 (a q a q a b c d q) q a a aq b aq c aq d q (a b q) q (+1)/ (q q) (abq q ) (a q) q ( 1)/ ( c) (c q q) a Poof Let d 0 i the Bailey pai at (47) to get the pai aq (aq aq/bc aq/bd aq/cd q) (35) bcd (aq/b aq/c aq/d aq/bcd q) (aq bq q ) (q abq q ) (Adews [1]) (36) (c/a q) (c q) (Heie [8]) (37) α 1 aq4 1 a a q ( 1) q q q (38) 19

20 α 1 0 β Substitute this pai ito (5) afte settig y a/z to get (a/z z q) q (+1)/ (aq q ) (q q) q ( 1)/ (aq q ) (q q) with espect to a a (zq aq/z q) (aq q q) (zq aq/z q) (aq q q) 0 (zq aq/z q ) (aq q q ) 1 aq 4 1 a (aq q q ) (q z aq /z q ) (a a/z z q ) ( 1) q + (zq aq /z q q ) The ext-to-last equality follows fom (35) (eplace q with q let d set b a/z ad c z) ad (36) follows upo eplacig a with ab ad z with b Next substitute the pai at (38) oce agai ito (5) this time settig y aq This gives ( aq aq/z q) (z q) q ( 1)/ aq ( aq q) (q q) z (aq aq/z q) 0 1 aq 4 1 a ( aq aq/z q) (aq (aq aq/z q ) aq/z q) (aq /z aq/z q ) ( aq/z q) ( aq q) (a z zq q ) q + (aq /z aq 3 /z q q ) a z q The ext-to-last equality follows oce agai fom (35) (eplace q with q let d set b z ad c zq) ad (37) follows upo eplacig aq with c ad z with a 3 A paticula case of the Bailey Tasfom I If we set y aq ad z aq i (5) the followig idetity esults povidig both seies covege: (aq q ) ( 1) 1 β (aq q ) ( 1 q) ( 1) α (39) This paticula case was ot used by Slate [17] ad may possibly be egaded as beig of lesse impotace fo two easos Fistly it is cetaily the case the both seies above will ot covege fo all Bailey pais (α β ) ad secodly eve if a seies-poduct idetity does esult the powe of q o the seies side may ot be quadatic i the expoet (ad may do ot coside such idetities as beig of Roges-Ramauja-Slate type) o else the esultig idetity is a easy cosequece of a moe geeal idetity Howeve we believe such idetities ae sufficietly iteestig to iclude some examples Coollay ( q q ) 1 q (q q ) ( q q ) (q q ) ( q 6 q 10 q 16 q 16 ) (310) 1 (q q ) q ( q q ) q 8 (q 4 q 4 ) (311) ( q q ) q (q q ) +1 ( q q ) (q q ) ( q q 14 q 16 q 16 ) (31) (q 3 q 3 ) ( q) (q q ) +1(q q) (q q ) (q q ) (q 18 q 18 ) (q 9 q 18 ) (313) Poof Fo the fist thee idetities above iset the Bailey pais at (6) (64) ad (66) espectively ito (39) (otig that a 1 q q espectively) ad i each case the esult follows afte some elemetay maipulatio of the esultig idetity Fo (313) substitute the Bailey pai at (8) ito (39) set a q ad simplify the esultig idetity 0

21 33 A paticula case of the Bailey Tasfom II If we set y q a ad let z i (5) the followig idetity esults: (q a q) a q ( 1)/ β (q a q) (aq q) (1 aq ) a q ( 1)/ α (314) This tasfomatio also gives ise to a umbe of iteestig false theta seies idetities ad we give seveal examples below See 35 below fo a explaatio of false theta seies Coollay 37 ( +1 )(q 3 q 3 ) ( 1) q (+1)/ (q q) + ( 1) q + ( q q) 0 (q q) +1( q q ) ( 1) q ( +)/ (q q) + 0 q 9 +3 ( 1+6 ) + q 10 + ( 18+9 ) q ( 4+1 ) 0 ( q q) q (+1)/ ( 1) (q q ) +1 0 q ( 16+8 ) + 0 q ( 6 3 ) (315) q ( 1 ) (316) 0 q ( 8+4 ) (317) ( 1) q + (318) Poof Let a q i the Bailey pai at (7) substitute the esultig pai ito (314) (with a q ) ad (315) follows afte simplifyig the left side ad e-idexig two of the esultig seies o the ight side Fo (316) let e ad set d q 3/ i the Bailey pai at (5) substitute the esultig pai (Slate s pai I(4)) ito (314) (with a 1) eplace q with q ad simplify Fo (317) substitute pai (61) ito (314) (with a 1) ad eaage To get (318) let a 1 ad set c q 1/ ad b q 1/ i Slate s geeal Bailey pai at (4) substitute the esultig pai (Slate s pai H(1) coected) ito (314) (agai with a 1) ad eaage the esultig idetity Remak 38 The tasfomatio at (314) does lead to idetities of Roges-Ramauja type but those we foud wee eithe ot ew o wee simple liea combiatios of existig idetities 34 Miscellaeous Idetities of the Roges-Ramauja-Slate type Fially we exhibit some idetities that follow fom Bailey pais ot listed by Slate pais that do follow howeve fom specializig the paametes i ou geeal Bailey pais We use two cases of the tasfomatio at (5) which wee used by Slate Fistly let y z to get a 1 q β (a q) (aq q) Secodly let z set y aq ad the eplace a with a ad q with q to get Coollay ( aq q ) a q β (a q ) ( aq q ) (a q q ) a q α (a q) (319) a q α (a q ) (30) q ( 1) (1 + q 1 )( 1 q) 1(q q) 1 + (q4 q 4 ) ( 1 q) (31) (q q ) q ( 1) (+1)/ ( 4 )( 1 q ) 1( q q ) (q q3 q 4 ) ( 1 q q 4 ) 1 + ( q 4 q 1 q 16 q 16 ) (3) 1

22 q ( +)/ ( 1) ( +1 )(q q) ( q 3 q 5 q 8 q 8 ) (33) q ( +3)/ ( 1) ( +1 )(q q) ( q q 7 q 8 q 8 ) (34) 1 + q 3 ()( ) q +3 ( q q ) 1 1 (q q) + (q q) ( q 18 q 30 q 48 q 48 ) + q 3 ( q 10 q 38 q 48 q 48 ) q 6 ( q q 46 q 48 q 48 ) q 3 ( q 6 q 4 q 48 q 48 ) (35) Poof Let d q ad a 1 i the Bailey pai at (47) The idetity at (31) follows upo substitutig the esultig pai ito (319) ad simplifyig Fo (3) set d q ad a i i the pai at (47) substitute the esultig pai ito (30) eplace q with iq ad eaage If we set a q ad d q i the pai at (48) we get (33) whe this pai is iseted i (30) afte eplacig q with q 1/ The idetity at (34) follows upo isetig the pai fomed by settig a q ad d q i the Bailey pai at (49) ito (30) ad the oce agai eplacig q with q 1/ Let v ad set u iq 3/ i the Bailey pai at (59) This gives the Bailey pai (with espect to a q ) α 0 β 0 1 α q8+3 q q 3 α q8+3 q q8+5 q q q 3 α α q8 3 q q 3 β q (1 + q)(1 + q )( q q ) 1 (1 + q 3 )(q 3 q) Note that the expessio fo β follows fom the fact that with the stated values fo u ad v K 8() (1 + q) 1 + q q (1 + q 3 ) (1 + q ) The idetity at (35) follows upo isetig this Bailey pai i (319) ad usig the Jacobi Tiple Poduct idetity to sum pais of seies 35 Theta-False Theta hybid idetities Roges [13] efeed to a seies of the fom (±1) q +s (±1) q +s + 1 (±1) q s (36) as a theta seies of ode whee Q + ad s Q Due to the Jacobi tiple poduct idetity the seies (36) may be expessed as the ifiite poduct q s q s q q Roges also took iteest i seies of the fom (±1) q +s 1 (±1) q s (±1) q +s ( ( s)+( s) ) (37)

23 which he called a theta seies of ode False theta seies ae ot epesetable as ifiite poducts but oetheless may have epesetatios as Roges-Ramauja type q-seies ad Roges gave a umbe of examples of idetities of false theta seies See [10 5 p 35 ff] fo futhe discussio ad umeous examples of false theta seies idetities Hee we give seveal examples of idetities whee oe side is a basic hypegeometic seies ad the othe side compises a theta poduct multiplied by a false theta seies We believe this type of idetity is ew Coollay 310 ( q q ) q ( +1 )(q q q ) ( q q ) (q q ) q ( 8+4 )! (38) P q + 0 q6 + ( 8+4 ) (39) ( +1 )(q q q) (q q) q + ( +1 )(q q q) ( q q) q ( +)/ ( +1 )(q q q) ( q q) (q q) ( q q) q ( +3)/ ( +1 )(q q q) ( q q) (q q) P 0 q6 +4 ( 4+ ) (q q) (330) 0 0 q 4 + ( 6+3 ) (331) q 4 +3 ( +1 ) (33) q + ( q q ) ( +1 )(q q ( 1) q 6 +4 ( 4+ ) (333) ) 0 " ( q q ) +1q ( q q ) # 1 + q + q + q 3 + q (q q ) (334) (q q ) (q q ) +1 (q q ) 0 Poof The idetity at (38) follows upo settig d q ad a q i the pai at (48) substitutig the esultig pai (Slate s pai I(10)) ito (314) (with a q) eplacig q with q ad fially q with q The idetity at (39) follows fom settig d q ad a q i the Bailey pai at (48) ad substitutig the esultig pai ito (319) while the idetity at (331) follows fom substitutig the same pai i (30) ad eplacig q with q 1/ Likewise the idetity at (330) follows as a cosequece settig d q ad a q i the Bailey pai at (49) ad substitutig the esultig pai ito (319) while fo (333) we poceed similaly afte ceatig a Bailey pai by istead settig a q (ad keepig d q ) The idetity at (33) follows upo isetig the pai used i the poof of (330) ito (30) afte eplacig q with q 1/ Fo (334) iset Slate s pai H(1) (a q c q ad b 0 i (4)) ito (314) (with a q) eplace q with q ad eaage 4 Cocludig Remaks I a pape by the secod autho [15] it was show that moe tha half of Slate s idetities could be deived fom just thee geeal Bailey pais togethe with seveal limitig cases of Bailey s lemma ad a associated family of q-diffeece equatios Hee we attempt to put Slate s wok i a boade cotext via geeal Bailey pais but without the use of q-diffeece equatios Both appoaches have thei meits ad both yield ew idetities It may well be woth exploig the combiatoial cosequeces of the ew idetities peseted hee We also ote that we have give just a sample of the idetities that may be poduced by the methods used i Coollaies ad that it is likely that may simila idetities may be poduced by employig othe Bailey pais 3

24 Refeeces [1] G E Adews O the q-aalog of Kumme s theoem ad applicatios Duke Math J 40 (1973) [] G E Adews Applicatios of basic hypegeometic fuctios SIAM Rev 16 (1974) [3] G E Adews R Askey ad R Roy Special fuctios Ecyclopedia of Mathematics ad its Applicatios 71 Cambidge Uivesity Pess Cambidge 1999 xvi+664 pp [4] G E Adews ad B C Bedt Ramauja s lost otebook Pat II Spige New Yok 009 xii+418 pp [5] W N Bailey Seies of hypegeometic type which ae ifiite i both diectios Quat JMath7 (1936) [6] W Chu Bailey s vey well-poised 6ψ 6-seies idetity J Combi Theoy Se A 113 (006) [7] G Gaspe ad M Rahma Basic hypegeometic seies d editio Ecyclopedia of Mathematics ad its Applicatios vol 96 Cambidge Uivesity Pess Cambidge 004 xxvi+48 pp [8] E Heie Utesuchuge übe die Reihe 1 + (α )( β ) ()( γ ) x + (α )( α+1 )( β )( β+1 ) x + ()( )( γ )( γ+1 ) J Reie Agew Math 34 (1847) [9] J McLaughli ad A V Sills Ramauja-Slate type idetities elated to the moduli 18 ad 4 J Math Aal Appl 344 (008) [10] J McLaughli A V Sills ad P Zimme Roges-Ramauja-Slate Type Idetities Electoic J Combiatoics 15 (008) #DS15 59 pp [11] Padmavathamma Some Studies i Patitio Theoy ad q-seies PhD thesis Uivesity of Mysoe Mysoe 1988 [1] S Ramauja The Lost Notebook ad Othe Upublished Papes Naosa New Delhi 1988 [13] L J Roges O two theoems of combiatoy aalysis ad some allied idetities Poc Lodo Math Soc 16 (1917) [14] H S Shukla A ote o the sums of cetai bilateal hypegeometic seies Poc Cambidge Philos Soc 55 (1959) 6 66 [15] A V Sills Idetities of the Roges-Ramauja-Slate type It J Numbe Theoy 3 (007) [16] L J Slate A ew poof of Roges s tasfomatios of ifiite seies Poc Lodo Math Soc () 53 (1951) [17] L J Slate Futhe idetities of the Roges-Ramauja type Poc Lodo MathSoc () 54 (195)