Modular Relations for the Sextodecic Analogues of the Rogers-Ramanujan Functions with its Applications to Partitions

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1 America Joural of Mathematical Aalysis 0 Vol. No. 6- Available olie at Sciece ad Educatio Publishig DOI:0.69/ajma--- Modular Relatios for the Sextodecic Aalogues of the Rogers-Ramauja Fuctios with its Applicatios to Partitios Adiga Chadrashekar * Nasser Abdo Saeed Bulkhali Departmet of Studies i Mathematics Uiversity of Mysore Maasagagotri Mysore Idia *Correspodig author: c_adiga@hotmail.com Received Jue 06 0; Revised Jue 9 0; Accepted July 0 0 Abstract I his Ph.D. thesis C. Gugg cosidered four fuctios of order 6 that are aalogues of the Rogers- Ramauja fuctios ad established modular relatios ivolvig these fuctios. I this paper we obtai 6 ew modular relatios for these fuctios. Furthermore we give partitio theoretic iterpretatios for some of our modular relatios. Keywords: Rogers-Ramauja fuctios theta fuctios partitios colored partitios modular relatios Cite This Article: Adiga Chadrashekar ad Nasser Abdo Saeed Bulkhali Modular Relatios for the Sextodecic Aalogues of the Rogers-Ramauja Fuctios with its Applicatios to Partitios. America Joural of Mathematical Aalysis vol. o. 0: 6-. doi: 0.69/ajma---.. Itroductio Throughout the paper we assume <. We use the stadard otatio 0 a ; : a ; : a j0 ad a; : a. 0 Ramauja s geeral theta fuctio is defied by f a b : + / / a b ab <. The Jacobi triple product idetity [[] Etry 9] i Ramauja s otatio is f a b a; ab b; ab ab; ab. Ramauja defied the followig three special cases of [[] Etry ]: ad : f ; ; + / ; 0 ; ψ : f / f : f ; 5 j For coveiece we defie f : f ; for a positive iteger. The well-kow Rogers-Ramauja fuctios are defied by ad G : 0 ; + H : 6 0 ; These fuctios satisfy the famous Rogers-Ramauja idetities ad G ; 5 ; 5 H. 5 5 ; ; I [] Ramauja remarks I have ow foud a algebraic relatio betwee G ad H viz.: 6 H { G } G { H } + GH { }. Aother oteworthy formula is HG GH.

2 America Joural of Mathematical Aalysis Each of these formulae is the simplest of a large class." I his lost otebook [8] Ramauja recorded forty beautiful modular relatios ivolvig the Rogers- Ramauja fuctios without proofs. The forty idetities were first brought before the mathematical world by B. J. Birch [8]. May of these idetities have bee established by L. J. Rogers [0] G. N. Watso [] D. Bressoud [0] A. J. F. Biagioli [] ad B. C. Berdt et al. [6] offered proofs of 5 of the 0 idetities. Recetly i Chapter 8 of their book [] G. E. Adrews ad Berdt collected proofs for all forty idetities. Most likely these proofs might have give by Ramauja himself. A umber of mathematicias tried to fid ew idetities for the Rogers-Ramauja fuctios similar to those which have bee foud by Ramauja [8] icludig Berdt ad H. Yesilyurt [5] Yesilyurt [6] S. Robis [9] ad C. Gugg []. Two beautiful aalogues of the Rogers-Ramauja fuctios are the Göllitz-Gordo fuctios which are defied as ad ; ; ; S : ; ; ; T :. Idetities 8 ad 9 ca be foud i L. J. Slater s list []. S.-S. Huag [6] has established a umber of modular relatios for S ad T. S.-L. Che ad Huag [] have derived some ew modular relatios ivolvig S ad T. N. D. Baruah J. Bora ad N. Saikia [] offered ew proofs of may of these modular relatios as well as establishig some ew relatios. E. X. W. Xia ad X. M. Yao [5] offered ew proofs of some modular relatios established by Huag [6] ad Che ad Huag []. They also established some ew relatios that ivolve oly Göllitz-Gordo fuctios. I view of the Ramauja s forty idetities may of the Rogers-Ramauja type fuctios were studied by may mathematicias. For example septic aalogues of the Rogers-Ramauja fuctios were studied by H. Hah [ 5] oic aalogues of the Rogers-Ramauja fuctios were studied by Baruah ad Bora [] cubic fuctios were studied by C. Adiga et al. [59] aother cubic fuctios were studied by K. R. Vasuki G. Sharath ad K. R. Rajaa [] dodecic aalogues of the Rogers- Ramauja fuctios were studied by Baruah ad Bora [] Robis [9] ad C. Gugg [] aother dodecic aalogues of the Rogers-Ramauja fuctios were studied by Vasuki ad P. S. Guruprasad [] Adiga Vasuki ad B. R. Srivatsa Kumar [0] established modular relatios ivolvig two fuctios of Rogers- Ramauja type the authors have studied two fuctios of order te [] ad more recetly Adiga et al. [6 8] have studied four fuctios of order fiftee. Almost all of these fuctios which have bee studied so far ca be foud i Slater s list []. I Chapter of his Ph.D. thesis C. Gugg [] cosidered the followig four sextodecic aalogues of the Rogers-Ramauja fuctios: ad ; f I : ; f ; f J : ; f 6 6 ; f K : ; f ; f L :. ; f 0 Gugg [] established twelve modular relatios ivolvig the fuctios I J K ad L icludig the followig two beautiful idetities: 6 f f f f f I I J J K K L L 6 f f II + JJ+ KK + LL. 5 f f f To prove his results Gugg [] applied a theorem of R. Blecksmith J. Brillhart ad I. Gerst [9] ad also employed the method give by Bressoud i his thesis [0]. The mai purpose of this paper is to establish several modular relatios that are aalogues of Ramauja s forty idetities ivolvig I J K ad L. May of these idetities that we fid have partitio theoretic iterpretatios. I the last sectio we extract partitio theoretic iterpretatios for some of these modular relatios. To prove our results we use the idea of Watso [] which he has used to prove some of Ramauja s forty idetities.. Mai Results I this sectio we preset a list of ew modular relatios for the fuctios I J K ad L which we establish i Sectio. For simplicity we use the otatios I : I J : J K : K ad L : L for a positive iteger. 6 8 f I + J + K + L f 6 f6 6 f f5 II 5+ JJ5+ KK 5+ LL 5 f f f5 f0 I L J K + K J L I f f ff f6f8 IJ JL KI LK 9 8 f f I L J K + K J L I f 0 f f

3 8 America Joural of Mathematical Aalysis f f60 I5L + J5K K5J + L5I f f f5 f0 f f I L + J K K J L I f f f 8 f f IK + JI KL+ LJ f ff6 I5I + J5J + K5K + L5L 5 5 f8 f60 f f 0 f f5 f f0 f0 f f8 f60 II + JJ + KK + LL 5 5 f8 f56 9 f f f f f f8 f f f8 f56 II5 + JJ5 + KK5 + LL5 5 5 f8 f0 f f 0 f5 f f f0 f0 f f8 f0 II 9 + J9J+ KK 9 + LL f8 f5 f f 50 f f9 f f6 f50 f f8 f5 6 0 IJ + JL + KI LK f f f8 f f f + f f f f f f 0 IK + JI KL + LJ f f f8 f f f f f f f f f I0J6 + J0L6 + K0I6 L0K6 f f5 f60 f f f 0 + f6 f0 f f0 f f5 IJ JL KI 6 0 LK 6 0 f f5 f60 f f f 0. f6 f0 f f0 f f Idetity 8 is the corrected versio of idetity.. foud i []. The followig two idetities are relatios ivolvig some combiatios of the fuctios defied i 0 ad the Göllitz-Gordo fuctios S ad T: KL f8 IJ f S f f IK+ JL f f8. T f f. Some Prelimiary Results The fuctio f ab satisfies the followig basic properties []: ad if is a iteger f ab f ba f a f aa 5 f a / / f a b a b f a ab b ab. Lemma. We have 5 f f f f f ψ f ff ad ψ. f f This lemma is a coseuece of the Jacobi triple product idetity ad Etry of []. The followig idetity follows easily from Etry foud i [] ψ Usig 8 oe ca easily establish the followig lemma: Lemma. We have a b a b 8 8 { ψ ψ } b a b a b b a. 9 The followig lemma ca be foud i [ Etry 0ii ad iii]: Lemma. We have f a b + f a b f a b ab 0 5 f a b f a b af b / a a b. Lemma. Let m [ s s r ] / l ms r r k ms r + s ad h mr m m s r / 0 r < s. Here [ x ] deote the largest iteger less tha or eual to x. The r s m h l k f f. For a proof of Lemma. see [].. Proofs of the Mai Results We prove our mai results usig ideas similar to those of Watso []. I all proofs oe expresses the left sides of the idetities i terms of theta fuctios by usig 0 ad. After clearig fractios we see that the right side ca be expressed as a product of two theta fuctios say with summatios idices m ad. Oe the tries to fid a chage of idices of the form

4 America Joural of Mathematical Aalysis 9 or αm+ β M ad γm+ δ N + b αm+ β 8M ad γm+ δ 8N + b αm+ β 6M ad γm+ δ 6 N + b so that the product o the right side decomposes ito the reuisite sum of two products of theta fuctios o the left side. Proof of. Usig 0 ad Lemma. we see that is euivalet to ψ f f 9 + f f f f f f. We have ψ f f 6 m m + +. m I this represetatio we make the chage of idices by settig m+ M ad m+ N + b where a ad b have values selected from the set {0 ± }. The m M N + a b / ad M + N + a+ b /. It follows easily that a b ad so m M N ad M + N where a. Thus there is oe-tooe correspodece betwee the set of all pairs of itegers m < m < ad triples of itegers MNa < M N < a. From we fid that ψ 6 a a M M + + am M N 8 N + + an N a f a a 9 a f f f + f f which is same as. Proof of5. Usig 0 ad Lemma. we see that 5 is euivalet to ψ f f f f f f f f. We have ψ f f 5 m m + +. m I these represetatios we make the chage of idices by settig m+ 8M ad m+ 8 N + b where a ad b have values selected from the set {0 ± ± ± }. The m M N + a b / 8 ad M + N + a+ b / 8. It follows easily that a b ad so m M N ad M + N where a. Thus there is oe-tooe correspodece betwee the set of all pairs of itegers m < m < ad triples of itegers MNa < M N < a. From 5 we fid that ψ a a M 56 M + + am M N 6 N + + an N a f a 6 f 9a 9 a f f f f + f f + f f + + f f

5 0 America Joural of Mathematical Aalysis which is same as. The proofs of the idetities 6 are very similar to those above so we omit the details. Proof of. Usig 0 ad Lemma. we see that is euivalet to f f f f f f f f ψ ψ. 6 Now chagig to ad the applyig Lemma. i the resultig idetity we may rewrite 6 i the form 5 5 { } f f f f. Thus we eed oly to establish. We have f f 5 8 m m +. m I these represetatios we make the chage of idices by settig m+ 6M ad 5 m+ 6 N + b where a ad b have values selected from the set {0 ± ± ± ± ± 5 ± 6 ± 8}. The m M N + a b /6 ad 5 M + N + 5 a+ b /6. It follows easily that a b ad so m M N ad 5M + N where a 8. Thus there is oeto-oe correspodece betwee the set of all pairs of itegers m < m < ad triples of itegers MNa < M N < a 8. From 8 we fid that 8 5 a M 0M + 0aM a M 6 N N + an N 8 a f 0+ 0a 00a a f 6 6a f f f f + f f + f f f f 9 Usig the same chage of idices for the product 5 we fid that 5 8 a a a M 0M + 0aM M N 6N N N 8 a a 0+ 0a 00a f a f 6 6a 9 f 0 50 f 0 6 f 60 0 f f f f f

6 f f + f f f f f f f f f f f f America Joural of Mathematical Aalysis 50 Subtractig 50 from 9 we deduce the desired result. I a similar way oe ca prove the idetities 5. Proof of ad. Usig 0 5 f Lemma. ad S ψ f T ψ we see that ad are euivalet respectively to ad ad ad f f f 5 f f ψ 9 f f f f f. ψ Applyig Lemma. we obtai 6 0 ψ + ψ f ψ ψ f. We may rewrite 5 ad 5 i the form { + } 8 0 f f f f ψ ψ ψ { } f f f f ψ ψ ψ Thus to establish ad it is suffices to prove 5 ad 5. We have ψ ψ f f m 8 m m m I this represetatio we make the chage of idices by settig m+ M ad m+ N + b where a ad b have values selected from the set {0 ± }. The m M N + a b / ad M + N + a+ b /. It follows easily that a b ad so m M N ad M + N where a. Thus there is oe-tooe correspodece betwee the set of all pairs of itegers m < m < ad triples of itegers MNa < M N < a. From 55 we fid that ψ a ψ a M 6 M ++ 8 am M N 6 N an N a f + 8a 88a a f + 8a 08a f f + f f + f f. We make the same argumet for the product ψ ψ to fid that a a ψ ψ a M 6 M ++ 8 am M N 6 N an N a a + 8a 88a f a + 8a 08a f f 8 f 6 f f + f f f f +.

7 America Joural of Mathematical Aalysis Usig the above two idetities we deduce the desired results. 5. Applicatios to the Theory of Partitios Some of our modular relatios yield theorems i the theory of partitios. I this sectio we preset partitio theoretic iterpretatios of some of our modular relatios. Defiitio 5. A positive iteger has k colors if there are k copies of available ad all of them are viewed as distict objects. Partitios of positive iteger ito parts with colors are called colored partitios". For example if is allowed to have two colors say r red ad g gree the all the colored partitios of are + r + g r + r + r r + r + g r + g + g ad g + g + g. It is easy to see that u v k ; is the geeratig fuctio for the umber of partitios of where all the parts are cogruet to umod v ad have k colors. For simplicity we defie r± s r sr s ; : ; where r ad s are positive itegers with r < s ad a a a; : a ;. i i Theorem 5. Let p deote the umber of partitios of ito parts cogruet to ± ± ± 5 ± mod6 where the parts cogruet to ± ± 5 mod6 have two colors. Let p deote the umber of partitios of ito parts cogruet to ± ± ± ± 6 8 mod6 where the parts cogruet to ± 6 8 mod6 have two colors. Let p deote the umber of partitios of ito parts cogruet to ± ± 5 ± 6 ± 8 mod6 where the parts cogruet to ± 6 8 mod6 have two colors. The for ay positive iteger p p p. Proof. Usig oe ca easily verify that is euivalet to ± ± ± 5± 5± ± 6 ; 56 ± ± ± 6± 6± 8 8 ; 6 ± 5± 6± 6± ± ; The three uotiets of 56 represet the geeratig fuctios for p p ad p respectively. Hece 56 is euivalet to p p p where we set p0 p0 p 0. Euatig coefficiets of o both sides yields the desired result. Example 5. The followig table illustrates the case 6 i Theorem 5. Table. Example for Theorem 5. p 6 8 p 6 9 p 5r + 6 r + 5 g + 6 g r + r + r + g + + g + g r g Theorem 5. Let p deote the umber of partitios of ito parts cogruet to ± ± ± 5 ± mod6 where the parts cogruet to ± ± mod6 have two colors. Let p deote the umber of partitios of ito parts cogruet to ± ± ± 5 ± 6 8 mod6 where the parts cogruet to ± 8 mod6 have two colors. Let p deote the umber of partitios of ito parts cogruet to ± ± ± 6 ± 8 mod6 where the parts cogruet to ± 8 mod6 have two colors. The for ay positive iteger p p p. Proof. Usig oe ca easily verify that is euivalet to ± ± ± 5± ± ± 6 ; ± ± ± 5± 6± 8 8 ; 6 + ± ± ± 6± ± ; 5 The three uotiets of 5 represet the geeratig fuctios for p p ad p respectively. Hece 5 is euivalet to p p + p where we set p0 p0 p 0. Euatig coefficiets of o both sides yields the desired result. Example 5.5 The followig table illustrates the case i Theorem 5.

8 America Joural of Mathematical Aalysis Table. Example for Theorem 5. p p 6 p + r r + r + g r + g r + r + r + r g + r + r + r + g r + + r + r + g + g g + + r + g + g + g g + g + g + g I a similar way oe ca prove the followig four theorems usig the modular relatios 6 8 ad 9 respectively: Theorem 5.6 Let p deote the umber of partitios of ito parts ot cogruet to ± ± 9 ± 6 ± ± mod 8 where the parts cogruet to ± g mod 8 have two colors ad ± 6 ± 8 mod 8 have three colors. Let p deote the umber of partitios of ito parts ot cogruet to ± 5 ± ± 5 ± 6 ± mod 8 where the parts cogruet to ± mod 8 have two colors ad ± 6 ± 8 mod 8 have three colors. Let p deote the umber of partitios of ito parts ot cogruet to ± ± 9 ± ± 6 ± 9 mod 8 where the parts cogruet to ± mod 8 have two colors ad ± 6 ± 8 mod 8 have three colors. Let p deote the umber of partitios of ito parts ot cogruet to ± ± ± 5 ± 6 ± mod 8 where the parts cogruet to ± mod 8 have two colors ad ± 6 ± 8 mod 8 have three colors. Let p 5 deote the umber of partitios of ito odd parts with two colors. The for ay positive iteger 6 p + p + p + p 6 p5. Theorem 5. Let p deote the umber of partitios of ito parts cogruet to ± ± ± 6 ± 8 ± 0 6 mod where the parts cogruet to ± ± 6 ± 0 6 mod have two colors ad ± 8 mod have three colors. Let p deote the umber of partitios of ito parts cogruet to ± ± ± 6 ± 8 ± 6 mod where the parts cogruet to ± ± 6 ± 6 mod have two colors ad ± 8 mod have three colors. Let p deote the umber of partitios of ito parts cogruet to ± ± ± 8 ± 0 ± 6 mod where the parts cogruet to ± ± 0 ± 6 mod have two colors ad ± 8 mod have three colors. Let p deote the umber of partitios of ito parts cogruet to ± ± 6 ± 8 ± 0 ± 6 mod where the parts cogruet to ± 6 ± 0 ± 6 mod have two colors ad ± 8 mod have three colors. Let p 5 deote the umber of partitios of ito parts cogruet to ± ± ± ± 5 ± 6 ± ± 9 ± 0 ± ± ± ± 5 mod. The for ay positive iteger 6 p + p + p + p 6 p5. Theorem 5.8 Let p deote the umber of partitios of ito parts cogruet to ± ± 5 ± ± 9 ± ± ± 9 ± mod 8. Let p deote the umber of partitios of ito parts cogruet to ± ± 5 ± ± 9 ± ± ± ± mod 8. Let p deote the umber of partitios of ito parts cogruet to ± ± ± ± ± 5 ± ± 9 ± mod 8. Let p deote the umber of partitios of ito parts cogruet to ± ± 5 ± ± ± 5 ± ± 9 ± mod 8. The for ay positive iteger p + p p + p. Theorem 5.9 Let p deote the umber of partitios of ito parts ot cogruet to ± ± 5 ± 6 ± 0 ± ± ± 6 ± 8 ± ± mod 8 where the parts cogruet to ± ± 9 ± 5 mod 8 have two colors ± ± 0 mod 8 have three colors ad ± mod 8 have four colors. Let p deote the umber of partitios of ito parts ot cogruet to ± ± ± 6 ± 0 ± ± 5 ± 6 ± ± 8 ± mod 8 where the parts cogruet to ± ± 9 ± mod 8 have two colors ± ± 0 mod 8 have three colors ad ± mod 8 have four colors. Let p deote the umber of partitios of ito parts ot cogruet to ± ± 6 ± ± 9 ± 0 ± ± 6 ± 8 ± ± mod 8 where the parts cogruet to ± ± 5 ± mod 8 have two colors ± ± 0 mod 8 have three colors ad ± mod 8 have four colors. Let p deote the umber of partitios of ito parts ot cogruet to ± ± ± 6 ± 0 ± ± ± 6 ± 8 ± 9 ± mod 8 where the parts cogruet to ± 9 ± 5 ± mod 8 have two colors ± ± 0 mod 8 have three colors ad ± mod 8 have four colors. Let p 5 deote the umber of partitios of ito parts ot cogruet to ± ± 8 ± ± 6 ± 0 mod 8 where the parts cogruet to ± ± 6 ± 0 ± ± 8 ± mod 8 have two colors. The for ay positive iteger 5 Refereces p + p + p p 5 + p5. [] C. Adiga B. C. Berdt S. Bhargava ad G. N. Watso Chapter 6 of Ramauja s secod otebook: Theta fuctios ad -series Mem. Amer. Math. Soc

9 America Joural of Mathematical Aalysis [] C. Adiga ad N. A. S. Bulkhali Some modular relatios aalogues to the Ramauja s forty idetities with its applicatios to partitios Axioms [] C. Adiga ad N. A. S. Bulkhali O certai ew modular relatios for the Rogers-Ramauja type fuctios of order te ad its applicatios to partitios Note di Matematica to appear. [] C. Adiga ad M. S. Surekha Some modular relatios for the Roger-Ramauja type fuctios of order six ad its applicatios to partitios Proc. J. math. Soc. to appear. [5] C. Adiga ad M. S. Surekha ad N. A. S. Bulkhali Some modular relatios of order six with its applicatios to partitios Gulf J. Math. to appear. [6] C. Adiga ad A. Vaitha New modular relatios for the Rogersâ Ramauja type fuctios of order fiftee Notes o Numb. Theor. Discrete Math [] C. Adiga ad A. Vaitha ad N. A. S. Bulkhali Modular relatios for the Rogers-Ramauja-Slater type fuctios of order fiftee ad its applicatios to partitios Romaia J. Math. Comput. Sci [8] C. Adiga ad A. Vaitha ad N. A. S. Bulkhali Some modular relatios for the Rogers-Ramauja type fuctios of order fiftee ad its applicatios to partitios Palestie J. Math [9] C. Adiga K. R. Vasuki ad N. Bhaskar Some ew modular relatios for the cubic fuctios South East Asia Bull. Math [0] C. Adiga K. R. Vasuki ad B. R. Srivatsa Kumar O modular relatios for the fuctios aalogous to Rogers-Ramauja fuctios with applicatios to partitios South East J. Math. ad Math. Sc [] G. E. Adrews ad B. C. Berdt Ramaujas Lost Notebook Part III Spriger New York 0. [] N. D. Baruah ad J. Bora Further aalogues of the Rogers- Ramauja fuctios with applicatios to partitios Elec. J. Combi. Number Thy. 00 # A05 pp. [] N. D. Baruah ad J. Bora Modular relatios for the oic aalogues of the Rogers-Ramauja fuctios with applicatios to partitios J. Number Thy [] N. D. Baruah J. Bora ad N. Saikia Some ew proofs of modular relatios for the Göllitz-Gordo fuctios Ramauja J [5] B. C. Berdt ad H. Yesilyurt New idetities for the Rogers- Ramauja fuctio Acta Arith [6] B. C. Berdt G. Choi Y. S. Choi H. Hah B. P. Yeap A. J. Yee H. Yesilyurt ad J. Yi Ramauja s forty idetities for the Rogers-Ramauja fuctio Mem. Amer. Math. Soc [] A. J. F. Biagioli A proof of some idetities of Ramauja usig modular fuctios Glasg. Math. J [8] B. J. Birch A look back at Ramauja s Notebooks Math. Proc. Camb. Soc [9] R. Blecksmith. J. Brillhat. ad I. Gerst. A foudametal modular idetity ad some applicatios. Math. Comp [0] D. Bressoud Proof ad Geeralizatio of Certai Idetities Cojectured by Ramauja Ph.D. Thesis Temple Uiversity 9. [] D. Bressoud Some idetities ivolvig Rogers-Ramauja-type fuctios J. Lodo Math. Soc [] S. L. Che ad S.-S. Huag New modular relatios for the Göllitz-Gordo fuctios J. Number Thy [] C. Gugg Modular Idetities for the Rogers-Ramauja Fuctio ad Aalogues Ph.D. Thesis Uiversity of Illiois at Urbaa- Champaig 00. [] H. Hah Septic aalogues of the Rogers-Ramauja fuctios Acta Arith [5] H. Hah Eisestei Series Aalogues of the Rogers-Ramauja Fuctios ad Parttitio Idetities Ph.D. Thesis Uiversity of Illiois at Urbaa-Champaig 00. [6] S.-S. Huag O modular relatios for the Göllitz-Gordo fuctios with applicatios to partitios J. Number Thy [] S. Ramauja Algebraic relatios betwee certai ifiite products Proc. Lodo Math. Soc. 90 p. xviii. [8] S. Ramauja The Lost Notebook ad Other Upublished Papers Narosa New Delhi 988. [9] S. Robis Arithmetic Properties of Modular Forms Ph.D. Thesis Uiver- sity of Califoria at Los Ageles 99. [0] L. J. Rogers O a type of modular relatio Proc. Lodo Math. Soc [] L. J. Slater Further idetities of Rogers-Ramauja type Lodo Math. Soc [] K. R. Vasuki ad P. S. Guruprasad O certai ew modular relatios for the Rogers-Ramauja type fuctios of order twelve Adv. Stud. Cotem. Math [] K. R. Vasuki G. Sharath ad K. R. Rajaa Two modular euatios for suares of the cubic-fuctios with applicatios Note di Matematica [] G. N. Watso Proof of certai idetities i combiatory aalysis J. Idia Math. Soc [5] E. X. W. Xia ad X. M. Yao Some modular relatios for the Göllitz-Gordo fuctios by a eve-odd method J. Math. Aal. Appl [6] H. Yesilyurt A geeralizatio of a modular idetity of Rogers J. Number Thy