Partial New Mock Theta Functions

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1 Alied Matheatics, (: 6-5 DOI:.59/j.a..9 Partial New Mock Theta Fuctios Bhaskar Srivas tava Deartet o Matheatics ad Astrooy, Luckow Uiversity, Luckow, Idia Abstract By usig a sile idetity o ie, I have roved sixty idetities coectig the ock theta uctios o Raauja with ock theta uctios recetly geerated by Adrews ad Briga et al, by takig their artial sus. Keywords Mock Theta Fuctio, Partial Mock Theta Fuctio, Q-Hyergeoetric Series. Itroductio I his last letter to G.H. Hardy[], Raauja wrote I discovered very iterestig uctios recetly which I call ock theta uctios. He also listed sevetee uctios our o order three, two grous ive i each grou o order ive ad three o order seve. He did ot exlai what he eat by order o the ock theta uctios. Later Watso gave three ore ock theta uctios o order three. Adrews while visitig Cabridge Uiversity oud otebooks o Raauja ad called the lost otebooks. The lost otebooks cotai six ock theta uctios ad Adrews ad Hickerso[7] called the o order six. The lost otebook also cotais eight idetities or the ock theta uctios ad Choi[] called these ock theta uctios o order te. Gordo ad McItosh[] gave eight ock theta uctios ad called the o order eight, though our o the are o lesser order. Hikai also gave a ock theta uctio ad called it o order two. Recetly Adrews i his aer[6] geerated soe ock theta uctios ad oud our o the iterestig. Briga, Hikai ad Lovejoy oud two ore ock theta uctios. We have ade corehesive study o these uctios i [6],[7] ad also their bilateral or i[8]. I soe o Raauja s uublished work, ow ublished as lost otebook, a uber o idetities ad exasio orulas or artial sus o theta uctios are etioed without roo. A study o these artial sus, idetities ad exasios has bee ade by Adrews ad he wrote that soe o these artial theta uctios idetities have iterestig uber theoretic iterretatios. These observatios otivated e to study the artial ock theta uctios. The deiitio o a artial ock theta uctios is that we take the artial su o the iiite series reresetig these uctios. Thus or ock theta uctio (, * Corresodig author: bhaskarsrivastav@yahoo.co (Bhaskar Srivastava Published olie at htt://joural.saub.org/a Coyright Scietiic & Acadeic Publishig. All Rights Reserved (, ( the artial ock theta uctio will be deied ad deoted as N, N (. ( Moreover, by takig the artial su, the additioal advatage is that covergece o the series is ot a restrait. I this aer we use a sile suatio idetity ad usig this idetity we coect the ock theta uctios o Adrews aog theselves ad the ock theta uctios o Briga et al aog theselves. Further we show the relatioshi betwee ock theta uctios o Adrews with the third, ith, sixth ad teth order ock theta uctio o Raauja ad with eighth order ock theta uctios o Gordo ad McItosh. The ock theta uctios o Adrews [6]: ad ad (, (. ( (, (. ( ( ( ; ( (, (. ( (. (. ( The ock theta uctios o Briga et al [9]:. Basic Results φ ( ( (.5 φ ( (. (.6

2 7 Alied Matheatics, (: 6-5 The ollowig - otatios have bee used For k <, k kj ( a; ( a, j k ( a, k kj ( a; ( a, j ( a ( a, k k k k ( a, a,..., a ; ( a ; ( a ;...( a ;. A geeralized basic hyergeoetric series with base is deied as a, a,..., a ; b, b..., b ;, z φ [ ] A A A A ( a...( aa z, z <. ( b...( ba (. Deiitios o Mock Theta Fuctios Third order ock theta uctios o Raauja:, ( : /(, Φ ( : /( ; ( : /(, (. γ ( : /( Fith order ock theta uctios o Raauja: ( : /(, F ( : /(, ( : F( :, ( (. Sixth order ock theta uctios o Raauja : L( : ( ( / ( Φ, ( L( : ( ( / (. To distiguish these uctios ro the others i otatio, we have ut a suix L i their sybols. Seveth order ock theta uctios o Raauja:. F ( : ( ; Eighth order ock theta uctios o Gordo ad McItosh: S ( ( : ( ;, ( ( ( : ( ; S, ( : ( ( / ( V V ( ( ;. ( Teth order ock theta uctios o Raauja:, X( : ( / ( (. χ( : ( / ( Secod order ock theta uctio ( ( ( ; ( (, A ( : ( ; ( ( (, B ( : ( ( µ ( :. ( ; Alterative deiitios o secod order ock theta uctios ( ( ( A ( ( φ, ; ;,, ( B ( ( φ, ; ;,, ( µ ( ( (, ; ;, ( φ.. Mai Suatio Theore We give a roo o the ollowig idetity Theore Proo αβ β α ( β β α. (. r r r r r r r The roo is give i y aer[5]. However it is very sile, so I reeat the roo here. It is a sile rearrageet o the series. ( β β αr ( β β α ( β β αr r r ( β β αr... ( β β αr r r

3 Bhaskar Srivastava: Partial New Mock Theta Fuctios 8 αβ αβ... α β β ( α α... α αβ β α which roves the theore., r r r r r 5. Relatios o Partial New Mock Theta Fuctios with Mock Theta Fuctios o Raauja 5.. Relatios aog Partial Mock Theta Fuctios o Adre ws ad Parti al Mock Theta Fucti os o Briga et al We give relatios betwee artial ock theta uctios o Adrews ad artial ock theta uctios o Briga et al. I the liit we have the ock theta uctios exressed as a series i artial ock theta uctios. By deiitio (, (, ( (5. ( ( (i Takig α /(, β i (., we have, ( ( ( ( [, (, ( ] ( (, (, ( (5. we have used (5. i the eultiate ste. (ii Lettig i (5., we have ( (, (, (. (5. I urther relatios the values o α ad β will be idicated i arethesis. (iii, (, ( (, ( ( ( (5. ( α /(, β / i (. ( (, (. (5.5 ( ( ( i (5. ( (v, (, ( ( ; ( ( (, ( ; /( ( /( ; ( (, ( ; (. (5.6 ( α, β i (. ( (. (5.7 ( i (5.6 (vii,,, φ ( ( φ ( ( φ (. (5.8 ( ( (viii φ φ φ, α, β i (. ( ( ( (. (5.9 ( i ( Relatios o Mock Theta Fuctios o Adrews wi th Third Order Mock Theta Fuctios o Raauja (i ( α (ii ( ; ( ;, ( ( /( ;, β ( ; /( ( ; ( ; ( ( ( ( ( ( (5. i (. ( ( ( ( (5. ( i (5. (iii ( α (. ( ( ( ( ( ;, ( ( (. (5. ( ; β ( / (, ( ; ( / ( ; i ( ( ( (. (5. ( i (5. (v ( α ( γ ( ( ;, ( γ (. (5. ( ; β /( ; /(, ( ; i (. ( ( γ(. (5.5 ( i (5. (vii ( α ( ( Φ ( Φ (. (5.6, ( ; ( /(, β /( i (. (viii ( Φ (. (5.7 ( ( i (5.6 (ix ( α ( Φ ( Φ (. (5.8, ( ( /( ;, β /( ; i (. (x ( Φ (. (5.9 ( ( i (5.8

4 9 Alied Matheatics, (: Relatios o Mock Theta Fuctios o Adrews with Fith Order Mock Theta Fuctios o Raauja (i ( α (ii,,, ( ( /( ;, β /( i (., ( ( ( ( ( (5. ( ( ( (5. ( i (5. (iii, (, ( ( ( α (, (. (5. ( /(, β /( ; i (. ( (iv (, ( (5. ( ( i (5. ( (v, (, ( ( ( ; ( (, ( ( ( ( α /( ;, β ( ; /( ; ( ; (5. i (. ( ( (, ( (5.5 ( ( ( i (5. (vii ( α (viii ( F ( F (. (5.6,,, ( ; ( ; /(, β /( ;, ( ; ( ; i (. ( F ( F ( (5.7 ( i (5.6 (ix, (, ( ( ( α (x, (. (5.8 ( /( ;, β /( ; i (. ( (, ( (5.9 ( ( ( i (5.8 (xi ( α (, (, ( ( ; ( ( (, (. (5. ( ( ; /( ;, β ( /( ; ( i (. ( (xii ( ( ( ( ; ( (, (. (5. ( ( ; ( i (5. (xiii, ( F, ( ( ; ( α F, (. (5. ( ; /(, β i (. (xiv ( F ( ( ; ( i (5. /( F, ( ( ;. ( Relatios o Mock Theta Fuctios o Adrews with Sixth Order Mock Theta Fuctios o Raauja (i ( α i (. ( ( ( Φ (, ( ( L ( ΦL(. (5. ( ( / (, β ( / ( (ii ( i (5. (iii ( α i (. ( ( ΦL(. (5.5 ( ( ( ( (, ( L ( L(. (5.6 ( ( ( ( / (, β ( / ( ( ( (. (5.7 ( i (5.6 ( 5.5. Relatios o Mock Theta Fuctios o Adrews with Seveth Order Mock Theta Fuctios o Raauja (i, ( ( F, ( ( L ( ( F, ( (5.8 (

5 Bhaskar Srivastava: Partial New Mock Theta Fuctios 5 ( α (ii ( /(, ( /( i (. β ( ( F ( ( ( ( F, ( (5.9 ( ( i ( Relatios o Mock Theta Fuctios o Adrews with Eighth Order Mock Theta Fuctios o Gordo ad Mc Itosh (i ( α i (. ( ( ( S (,, ( ; ( ( ( S, ( (5. ( ( ; ( ( /(, β ( /( ; ( ( ( (ii ( S, ( (5. ( ( ; ( i (5. (iii, ( V, ( ( ; ( α i (., ( ; ( /(, β ( V ( (5. /( ; ( ; (, ( ; ( V ( (5. ( i (5. (v ( α (. ( ( S (,, ( ( S, ( (5. ( ( /( ;, β, ( /( i ( ( S (. (5.5 ( i (5. (vii ( α (. ( ( S (,, ( ( ( S, (. ( ( (5.6 ( ; /( ;, β /( ( i ( (viii ( S, ( ( ( (5.7 ( i ( Relati os o Mock Theta Fuctios o Adrews with Teth Order Mock Theta Fuctios o Raauja ( (i, ( ( X ( ( α ( ( X( (5.8 ( / (, β ( i (. (ii ( ( ( X(. (5.9 ( i (5.8 (iii ( α (, ( ( χ( ( ( χ( (5.5 ( ( / (, β ( i (. ( ( ( χ(. (5.5 ( i ( Relati os o Mock Theta Fuctios o Adrews ad Mock Theta Fuctios o Briga et al with Secod Order Mock Theta Fuctios (i, φ ( ( ( /( ; µ ( ( ( ; [ ( ( ( ] µ ( (5.5 ( α ( ( /( ;, β ( ( ; ( ; i (. /( (ii φ( ( ( ; [ ( ( ( ] µ (. (5.5 ( i (5.5 (iii, φ ( ( ( ; µ ( ( ( ; [ ( ] µ ( (5.5 ( α ( ( /( ;, β ( ( ; i (. φ ( ( ( ; [ ( ] µ ( (5.55 ( i (5.5 (v (, ( ; ( B( ( ( ; ( B ( (5.56 ( ;

6 5 Alied Matheatics, (: 6-5 ( α ( ; ( ; /( ; ( ; ( ;, β /( ( i (. ( ; ( ( B ( (5.57 ( ( ; ( i (5.56 ( ( ; (vii (, ( A ( ( ( ; ( ; ( ; 6 ( ( A ( (5.58 ( ; ( ; ( ; ( ( ( α ( ( ( ; i (. (viii ( ; ( ( ( 6 ( ; ( ; (, β ( ; ( ; ( ( A ( (5.59 ( i (5.58 (ix, ( ; ( B( ( ; ( ; ( ; [ ( ( ] B ( (5.6 ( ; ( ( α ( ; ( /( ( ;, β /( ( i (. ( ; [ ( ( ] (x ( B ( (5.6 ( ; ( ; ( ; ( i ( Coclusios By the hel o a sile suatio orula idetities coectig various artial ock theta uctios are give. The cosideratio o artial sus o the ock theta uctios is helul as there is o costrait about covergece o the series. These artial idetities ay be helul i givig uber theoretic iterretatios [] K. Alladi, A cobiatorial study ad coariso o artial theta idetities o Adrews ad Raauja, Raauja J. (, 7-. [5] G. E. Adrews, O basic hyergeoetric series, ock theta uctios ad idetities-i, Quart. J. Math. (Oxord 7 (966, 6-8. [6] G. E. Adrews, -orthogoal olyoials, Rogers-Raauja idetities, ad ock theta uctios, rerit. [7] G. E. Adrews, D. Hickerso, Raauja s lost otebook VII: The sixth order ock theta uctios, Adv. i Math. 89 (99, 6-5. [8] B. Berdt, B. Ki ad A. Yee, Raauja s lost otebook: cobiatorial roos o idetities associated with Heie s trasoratio or artial theta uctios. J. Cobi. Theory Ser. A 7 (, [9] K.Briga, K. Hikai, J. Lovejoy, O the odularity o the uiied WRT ivariats o certai Seiert aiolds, Adv.Al. Math., to aear. [] You-Seo Choi, Teth order ock theta uctios i Raauja s lost otebook IV, Tras. Aer. Math. Soc. 5( (, [] N.J. Fie, Basic Hyergeoetric series ad alicatio. Matheatical surveys ad Moograhs, Aer. Math. Soc. 7 (998. [] G. Gaser ad M. Raha, Basic Hyergeoetric Series, Cabridge Uiversity Press, Cabridge (99. [] B.Gordo ad R.J. McItosh, Soe eighth order ock theta uctios, J. Lodo Math. Soc. 6( (, -5. [] S. Raauja, Collected Paer, Cabridge Uiversity Press 97, rerited by Chelsea New York, (96. [5] Bhaskar Srivastava, Raauja s Mock Theta Fuctios, Math. J. Okayaa Uiv. 7 (5, 6-7. [6] Bhaskar Srivastava, A geeralizatio ad study o our ew ock theta uctios, (Couicated. [7] Bhaskar Srivastava, Soe ew ock theta uctios, acceted i Math. Sci. Research J. [8] Bhaskar Srivastava, A study o Bilateral ew ock theta uctios, (Couicated. [9] G.N. Watso, The ial roble: A accout o the ock theta uctios, J. Lodo Math. Soc. (96, [] G.N. Watso, The Mock Theta Fuctio II, Proc. Lodo Math. Soc. ( (97, 7-. REFERENCES [] R. P. Agarwal, Certai basic hyergeoetric idetities associated with ock theta uctios, Quart. J. Math.(Oxord (968, -8. [] R. P. Agarwal, Mock Theta Fuctios- a aalytical oit o view, Proc. Nat. Acad. Sci, 6 (99, [] K. Alladi, A artial theta idetity o Raauja ad its uber-theoretic iterretatio, Raauja J. (9, [] D. Zagier, Raauja s ock theta uctios ad their alicatios (ater Zwegers ad Oo-Briga, S eiaire Bourbaki. Vol. 7/8. Ast erisue No. 6 (9, Ex. No. 986, vii viii, 6 (.

A Study on Product Formulae for Mock Theta Functions of Different Order

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