A Study of a Generalization of Ramanujan s Third Order and Sixth Order Mock Theta Functions

Transcript

1 Applied Mathematics 01 (5): DOI: /j.am A Study of a Generalization of Ramanujan s Third Order and Sixth Order Mock Theta Functions Sameena Saba Karamat Husain Muslim Girls P.G. College Lucknow Abstract A generalization of sixth order and third order mock theta functions is given and shown that these generalized functions belong to the class of FF qq -functions. Multibasic expansion and q-integral representation of these generalized functions is also given. Keywords Basic Hypergeometric Series Mock Theta Functions q-integral Multibasic Expansions 1. Introduction S. Ramanu jan in his last letter to G.H. Hardy[9 pp ] introduced seventeen functions whom he called mock theta functions as they were not theta functions. He stated two conditions for a function to be a mock theta function: (0) For every root of unity ζζ there is a θθ-function θθ ζζ (qq) such that the difference ff(qq) θθ ζζ (qq) is bounded as qq ζζ radially. (1) There is no single θθ-function which works for all ζζ i.e. for every θθ-function θθ(qq) there is some root of unity ζζ for which difference ff(qq) θθ(qq) is unbounded as qq ζζ radially. Of the seventeen mock theta functions four were of third order ten of fifth order in two groups with five functions in each group and three of seventh order. Ramanujan did not specify what he meant by the order of a mock theta function. Later Watson[15] added three more third order mock theta functions making the four third order mock theta functions to seven. G.E. Andrews[1] while v isiting Trin ity College Cambridge University discovered some notebooks of Ramanujan and called it the Lost Notebook. In the Notebook Andrews found seven more mock theta functions and some identities and Andrews and Hickerson[] called them of sixth order. The sixth order mock theta functions of Ramanujan are Φ(qq) = ( 1 qq nn (qq; qq ( qq * Corresponding author: saba08084@gmail.com (Sameena Saba) Published online at Copyright 01 Scientific & Academic Publishing. All Rights Reserved Ψ(qq) = ( 1 qq (nn+1) (qq; qq ( qq+1 qqnn(nn+1) ( qq ρ(qq) = (qq; qq +1 σ(qq) = qq (nn+1)(nn+) ( qq (qq;qq +1 0 Mathematics Subject Classification.33D15 Key words and phrases. Basic Hypergeometric series mock theta functions q-integral multibasic expansions. λ(qq) = ( 1 qq nn (qq;qq ( qq and nn =0 μ(qq) = ( 1 (qq; qq ( qq γ(qq) = qqnn (qq (qq 3 ;qq 3. The third order mock theta functions of Ramanujan are qqnn ff(qq) = ( qq) nn qqnn φφ(qq) = ( qq ;qq qqnn ψψ(qq) = (qq; qq nn =1 qq nn χχ(qq) = (1 qq + qq ) (1 qq nn + qq nn ) ωω(qq) = qq nn (nn+1) (qq;qq +1

2 158 Sameena Saba: A Study of a Generalization of Ramanujan s Third Order and Sixth Order Mock Theta Functions qqnn(nn+1) νν(qq) = ( qq; qq ) nn+1 We give a generalization of the sixth order and the third order mock theta functions. The generalized sixth order mock theta functions are qq nn (nn+1) and ρρ(qq) = (1 + qq + qq ) (1 + qq nn+1 + qq 4nn+ ). Φ(tt αα zz; qq) = 1 Ψ(tt αα zz; qq) = 1 (tt) ρ(tt αα zz; qq) = 1 (tt) ( 1)nn (tt qq nn (nn 3)+nnnn zz nn zz qq ;qq nn nn =0 (1.1) ( zz qq ;qq ( 1)nn (tt qq nn (nn 1)+nnnn zz nn +1 zz qq ;qq nn ( zz qq ;qq +1 nn =0 (1.) (tt) nn (nn 3) (tt) +nnnn nn qq zz nn ( zz;qq (zz qq ;qq +1 nn =0 (1.3) and σ(tt αα zz;qq) = 1 (tt) λ(tt αα zz; qq) = 1 μ(tt αα zz; qq) = 1 γ(tt αα zz;qq) = 1 (tt qq nn (nn 1) +nnnn zz nn +1( zz qq ;qq +1 (zz qq;qq +1 nn =0 (1.4) (tt) (tt) (tt) (tt) ( 1)nn nn qq nnnn qq 3 zz ;qq nn ( qq zz;qq (1.5) (tt) ( 1)nn nn qq nn (αα 1) qq 3 zz ;qq nn ( qq zz;qq nn =0 (1.6) (tt qq nn (nn 3)+nnnn zz nn. (1.7) (νν zz;qq (νν 4 zz;qq For tt = 0αα = 1 we have the generalized functions of Choi[4]. For zz = qq ll and using induction these functions satisfy the Ramanujan s requirement for a mock theta function. The generalized third order mock theta functions are and where ff(tt αα ββ zz; qq) = 1 (tt) φφ(tt αα ββ zz;qq) = 1 ψψ(tt αα ββ zz;qq) = 1 νν(tt αα ββ zz; qq) = 1 (tt) ωω(tt αα ββ zz; qq) = 1 χχ(tt ββ zz; qq) = 1 ρρ(tt ββ zz; qq) = zz4 qq 4 1 (tt) (tt) (tt) (tt) (tt) For ββ = 1 and zz = qq we have generalized five third order mock theta functions namely ff φφ ψψ νν ωω of Andrews[1]. For tt = 0 ββ = 1 αα = qq and zz = qq the generalized functions ff φφ ψψ and χχ reduce to the third order mock theta functions of Ramanujan and ωω νν and ρρ to the third order mock theta functions of Watson[15]. In this study we will show that these generalized functions are FF qq -functions. This is done in section 3. (tt qq nn 4nn +nnnn αα nn zz nn nn =0 (1.8) ( zz;qq ( αααα qq ;qq (tt qq nn 3nn +nnnn zz nn (1.9) ( αα zz qq ;qq (tt qq nn nn +nnnn zz nn +1 (1.10) (ααzz qq ;qq +1 (tt qq nn nn +nnnn zz nn nn =0 (1.) ( αα zz qq 3 ;qq +1 (tt qq nn 5nn 4+nnnn αα nn zz 4(nn +1) nn =0 (1.1) (zz qq ;qq +1 ( αα zz qq 3 ;qq +1 (tt qq nn 3nn +nnnn zz nn (νννν ;qq ( νν zz ;qq nn =0 (1.13) (tt qq nn 3nn +nnnn zz 4nn. (1.14) (νν zz qq;qq +1 ( νν zz qq ;qq +1 νν = ee ππππ 3. Using the difference equation we derive relations between generalized sixth order mock theta functions and relation between generalized third order mock theta functions. This we do in section 4. In section 5 we give a q-integral representation and in section 6 we g ive mu ltibasic expansion for these generalized functions.. Notations

3 Applied Mathematics 01 (5): We shall use the following usual basic hypergeometric notations: For qq kk < 1 (aa; qq kk = (1 aa)(1 aaqq kk ) 1 aaqq kk(nn 1) nn 1 (aa; qq kk ) 0 = 1 (aa; qq kk ) = jj =0 (1 aaqq kkkk ) φφ aa 1 aa rr :cc 11 cc 1rr1 : :cc mm 1 cc mm rrmm bb 1 bb ss :ee 1 1 ee 1ss1 : :ee mm 1 ee ;qq qq 1 qq mm ; zz mm ssmm ( aa 1 aa rr ; qq zz (qq bb 1 bb ss ; qq ( 1 qq nn nn 1+ss rr nn = mm cc jj1 cc jjrrjj ; qq jj nn ss nn jj rr jj. nn ( 1 qq jj jj =1 ee jj1 ee jj ssjj ; qq jj nn AAφφ AA 1 [aa 1 aa aa AA ;bb 1 bb bb AA 1 ;qq 1 zz] = ( aa 1 ; qq 1 (aa AA ; qq 1 zz nn zz < 1. (bb 1 ; qq 1 (bb AA 1 ; qq 1 (qq 1 ;qq 1 3. The Generalized Functions are FF qq -Functions We show these generalized functions are FF qq - functions. Theorem 1 The generalized functions Φ(tt αα zz;qq) ψ(tt αα zz; qq) ρ(tt αα zz; qq) γ(tt αα zz; qq) σ(tt αα zz; qq) and the generalized functions ff(tt αα ββ zz; qq) φφ(tt αα ββ zz; qq) ψψ(tt αα ββ zz; qq) νν(tt αα ββ zz;qq) χχ(tt ββ zz; qq) ρρ(tt ββ zz; qq) ωω(tt αα ββ zz; qq) are FF qq - functions. Proofs We shall give the proof for Φ(tt αα zz; qq) only. The proofs for the other functions are similar hence omitted. Applying the difference operator DD qq tt to Φ(tt αα zz;qq)we have ttdd qq tt Φ(tt ααzz; qq) = Φ(tt αα zz; qq) Φ(tttt αα zz; qq) = 1 ( 1 (tt qq nn(nn 3)+nnnn zz nn (zz qq; qq (tt) ( zz qq;qq 1 ( 1 (tttt qq nn(nn 3)+nnnn zz nn (zz qq; qq (tttt) ( zz qq; qq = 1 ( 1 (tt qq nn(nn 3)+nnnn zz nn (zz qq; qq (tt) ( zz qq;qq) nn 1 ( 1 (tt qq nn(nn 3)+nnnn zz nn (zz qq; qq (1 ttqq nn ) (tt) ( zz qq; qq) nn = tt ( 1 (tt qq nn(nn 3)+nn(αα +1) zz nn (zz qq; qq (tt) ( zz qq; qq) nn = ttφ(tt αα + 1 zz; qq). So DD qq tt Φ(tt ααzz; qq) = Φ(tt αα + 1 zz; qq). Hence Φ(tt αα zz; qq) is a FF qq - function. As stated earlier the proofs for other functions are similar so omitted. 4. Relations between the Generalized Functions of Sixth Order Mock Theta Functions and Relations between Generalized Functions of Third Order Mock Theta Functions Theorem (i)φ(tt αα zz; qq) = 1 ( 1 (tt qq nn(nn 3)+nnnn zz nn (zz qq; qq (tt) ( zz + zz qq; qq+1 qq Ψ (tt αα zz; qq). (ii)σ(tt αα zz; qq) = zz 1 + zz qq DD qq ttρ(tt αα zz;qq). (iii)dd qq tt φφ(tt αα ββ zz; qq) = 1 + αα zz νν (tt αα ββ zz; qq). qq 3 (iv)ψψ tt αα qq ββ zz; qq = zzdd qq tt νν(tt αα ββ zz; qq). Proof of (i) Φ(tt αα zz; qq) = 1 ( 1 (tt qq nn(nn 3)+nnnn zz nn (zz qq;qq (1 + zz qq nn 1 ) (tt) ( zz qq;qq) nn +1 = 1 ( 1 (tt qq nn(nn 3)+nnnn zz nn (zz qq; qq (tt) ( zz qq; qq) nn +1 + zz 1 ( 1 (tt qq nn(nn 1)+nnnn zz nn+1(zz qq;qq qq (tt) ( zz qq; qq) nn+1 = 1 ( 1 (tt qq nn(nn 3)+nnnn zz nn (zz qq; qq (tt) ( zz + zz qq; qq +1 qq Ψ (tt αα zz;qq)

4 160 Sameena Saba: A Study of a Generalization of Ramanujan s Third Order and Sixth Order Mock Theta Functions which proves Theorem (i). Proof of (ii) zz 1 + zz qq σ(tt αα zz; qq) = ( tt qq (tt) nn =0 nn(nn 1) +nnnn zz nn ( zz; qq (zz qq; qq +1 = zz 1 + zz qq DD qq ttρ(tt αα zz;qq) which proves Theorem (ii). Proof of (iii) Writing αα for αα in φφ(tt αα ββ zz; qq) we have DD qq tt φφ(tt αα ββ zz; qq) = 1 ( tt qq nn nn+nnnn zz nn (tt) ( αα zz qq ; qq = 1 + αα zz ) nn qq 3 νν (tt αα ββ zz; qq) which proves Theorem (iii). Proof of (i v) Writing αα for αα and then αα for αα in ψψ(tt αα ββ zz; qq) in (1.8) we have qq ψψ tt αα zz ββ zz; qq = ( tt qq nn nn+nnnn zz nn qq (tt) ( αα zz qq 3 ; qq = zzzz ) qq tt νν(tt αα ββ zz; qq) nn+1 which proves Theorem (iv). 5. q-integral Representation for the Generalized Functions of Sixth and Third Order Mock Theta Functions The q-integral was defined by Thomae and Jackson[7 p. 19] as 1 ff(tt)dd qq tt = (1 qq) ff(qq nn ) qq nn. 0 We now give the q-integral representation for the generalized sixth order mock theta functions and also for generalized third order mock theta functions. Theorem 3(a) (i)φ(qq tt αα zz; qq) = ( 1 qq) 1 (qq; qq) tt 1 (wwww; qq) Φ(0 aaaa zz; qq)dd qq ww. (ii) Ψ(qq tt αα zz; qq) = ( 1 qq) 1 (qq; qq) (iii)ρ(qq tt αα zz; qq) = ( 1 qq) 1 (qq; qq) (iv)γ(qq tt αα zz; qq) = ( 1 qq) 1 (qq; qq) nn =0 tt 1 (wwww; qq) Ψ(0 aaaa zz;qq)dd qq ww. tt 1 (wwww; qq) ρ(0 aaaa zz;qq)dd qq ww. tt 1 (wwww; qq) γ(0 aaaa zz; qq)dd qq ww. (v) σ(qq tt αα zz; qq) = ( 1 qq) 1 (qq; qq) tt 1 (wwww; qq) σ(0 aaaa zz; qq)dd qq ww. PPPPPPPPPP We shall give the detailed proof for Φ(qq tt αα zz;qq). The proofs for the other functions are similar so omitted. Limiting case of qq-beta integral[7 p. 19 (1..7)] is 1 (qq xx ;qq) = ( 1 qq) 1 (qq ;qq) Now Φ(tt αα zz; qq) = 1 ( 1 (tt qq nn(nn 3)+nnnn zz nn (zz qq; qq (tt) ( zz qq; qq) nn Replacing t by qq tt and qq αα by a we have tt xx 1 (tttt; qq) dd qq tt. (5.1)

5 Applied Mathematics 01 (5): But and since qq αα = aa Hence Φ(qq tt αα zz; qq) = 1 ( 1 (qq tt qq nn(nn 3)+nnnn zz nn (zz qq; qq (qq tt ) ( zz qq; qq = ( 1 qq nn(nn 3)+nnnn zz nn (zz qq; qq ( zz qq;qq (qq nn+tt ) = ( 1 qq nn (nn 3)+nnnn zz nn (zz qq;qq (1 qq) 1 ( zz ww qq; qq (qq; qq+tt 1 (wwww; qq) dd qq ww by (5.1) = ( 1 qq) 1 (qq ;qq) by using (5.3) (5.) can be written as ww tt 1 ( 1)nn qq nn (nn 3) zz nn zz qq ;qq nn ( zz qq ;qq (wwww; qq) (aaaa) nn dd qq ww. (5.) Φ(0 αα zz; qq) = ( 1 qq nn (nn 3)+nnnn zz nn (zz qq; qq ( zz qq; qq Φ(0 aa zz;qq) = ( 1 (aa qq nn (nn 3) zz nn (zz qq;qq ( zz. qq;qq ( 1)nn (aaaa )nn qq nn (nn 3) zz nn zz qq;qq nn Φ(0 aaaa zz;qq) =. (5.3) Φ(qq tt αα zz; qq) = ( 1 qq) 1 (qq; qq) ( zz qq ;qq tt 1 (wwww; qq) Φ(0 aaaa zz; qq)dd qq ww which proves (i). The proofs for all other functions are similar. Theorem 3(b) The q-integral representation for the generalized third order mock theta functions: (i)ff(qq tt αα ββ zz; qq) = ( 1 qq) 1 (qq; qq) tt 1 (wwww; qq) ff(0 αα aaaa zz; qq)dd qq ww. (ii)φφ(qq tt αα ββ zz; qq) = ( 1 qq) 1 (qq; qq) tt 1 (wwww; qq) φφ(0 αα aaaa zz; qq)dd qq ww. (iii)ψψ(qq tt αα ββ zz; qq) = ( 1 qq) 1 (qq; qq) tt 1 (wwww; qq) ψψ(0 αα aaaa zz; qq)dd qq ww. (iv) νν(qq tt αα ββ zz; qq) = ( 1 qq) 1 (qq; qq) tt 1 (wwww; qq) νν(0 αα aaaa zz;qq)dd qq ww. (v)χχ(qq tt ββ zz; qq) = ( 1 qq) 1 (qq; qq) (vi)ρρ(qq tt ββ zz; qq) = ( 1 qq) 1 (qq; qq) tt 1 (wwww; qq) χχ(0 aaaa zz; qq)dd qq ww. tt 1 (wwww; qq) ρρ(0 aaaa zz; qq)dd qq ww. (vii)ωω(qq tt αα ββ zz;qq) = ( 1 qq) 1 (qq;qq) tt 1 (wwww;qq) ωω(0 αα aaaa zz; qq)dd qq ww. Proof The proofs are similar to given above for Φ(qq tt αα zz;qq) so the Theorem 3(b) follows. 6. Multibasic Expansions of Generalized Functions of Sixth and Third Order Mock Theta Functions Using the summation formula[7 (3.6.7) p. 71] and[8 Lemma 10 p. 57] we have the multibasic expansion

6 16 Sameena Saba: A Study of a Generalization of Ramanujan s Third Order and Sixth Order Mock Theta Functions 1 aapp kk qq kk 1 bb pp kk qq kk (aa bb;pp) kk (ccaa bbbb ;qq) kk qq kk mm =0 αα mm +kk = mm =0 αα mm. (6.1) (1 aa )(1 bb) (qqaaaa bb ;qq) kk (aaaa ccbbbbbb ;pp) kk Corollary 1 Letting qq qq and cc in (6.1) we have 1 aapp kk qq kk 1 bb pp kk qq kk (aa bb;pp) kk qq kk +kk (aaaa bbbb ;pp) mm (cccc aaaa bbbb ;qq) mm (qq aaaa bb;qq) mm (aaaa ccbbbbbb ;pp) mm mm =0 αα mm +kk = mm =0 αα mm. (6.) (1 aa )(1 bb) (qq aaqq bb;qq ) kkbb kk kk +kk pp Corollary Letting qq qq 3 and cc in (6.1) we have 1 aapp kk qq 3kk 1 bbpp kk qq 3kk 3kk +3kk (aa bb ;pp) kk qq (aaaa bbbb ;pp) mm qq mm +mm (qq aaqq bb ;qq ) mm bb mm mm +mm pp mm =0 αα mm +kk = mm =0 αα mm. (6.3) (1 aa )(1 bb) (qq 3 aaqq 3 bb ;qq 3 ) kkbb kk kk +kk pp Corollary 3 Letting qq qq 5 and cc in (6.1) we have 1 aapp kk qq 5kk 1 bb pp kk qq 5kk (aa bb ;pp) kk qq 5kk +5kk 3mm +3mm (aaaa bbbb ;pp) mm qq (qq 3 aaqq 3 bb ;qq 3 ) mm bb mm mm +mm pp mm =0 αα mm+kk = mm =0 αα mm. (6.4) (1 aa )(1 bb)(qq 5 aaqq 5 bb ;qq 5 ) kkbb kk kk +kk pp 5mm +5mm (aaaa bbbb ;pp) mm qq (qq 5 aaqq 5 bb ;qq 5 ) mm bb mm mm +mm pp In the following Theorem we now give multibasic expansions for generalized functions of sixth order mock theta functions and third order mock theta functions. We give detailed proof for Φ(tt aa zz;qq) and for other functions we write only the specialized parameters. Theorem 4(a) The expansions for generalized functions of sixth order mock theta functions. (i)φ(tt αα zz; qq) = 1 ( 1) kk (1 ttqq 4kk 1)(1 qq kk +)(tt; qq) kk 1 (zz qq; qq ) kk qq kk 3kk +kkkk zz kk (tt) (1 qq kk + )( zz qq; qq) kk qq 0:zz qq kk 1 0: ttqq 3kk qq 3kk +3 : qq kk +3 : zz qq kk 1 zz qq kk : 00: ; qq qq qq 3 ; zz qq αα. (ii)ψ(tt αα zz; qq) = 1 ( 1) kk (1 ttqq 4kk 1)(1 qq kk+)(tt;qq) kk 1 (zz qq; qq ) kk qq kk kk +kkkk kk +1 zz (tt) (1 qq kk + )( zz qq; qq) kk +1 qq 0:zz qq kk 1 0: ttqq 3kk qq 3kk +3 : qq kk +3 : zz qq kk zz qq kk+1 : 00: ; qq qq qq 3 ; zz qq αα. (iii)ρ(tt αα zz; qq) = 1 ( 1 ttqq 3kk 1)(1 qq kk+)(tt; qq) kk 1 ( zz; qq) kk qq (tt) (1 qq kk + )(zz qq;qq ) kk +1 qq zzqqkk : ttqq kk 1 qq kk + : qq kk +3 :zz qq kk +1 ;qq qq ;zzqq αα. 0: (iv)γ(tt αα zz; qq) = 1 ( 1 ttqq 4kk 1)(1 qq kk +)(tt; qq) kk 1 qq kk 3kk +kkkk zz kk (tt) (1 qq kk + )(νν zz;qq) kk (νν 4 zz;qq) kk kk 3kk +kkkk zz kk qq 000: ttqq 3kk qq 3kk +3 : qq kk +3 νν zzqq kk νν 4 zzqq kk : 00: ; qq qq3 ;zz qq αα. (v)σ(tt αα zz; qq) = 1 ( 1 ttqq 3kk 1)(1 qq kk+1)(tt; qq) kk 1 ( zz qq; qq) kk +1 qq kk kk +kkkk zz kk+1 (tt) (1 qq kk+1 )(zz qq; qq ) kk +1 Proof of (i) in (6.3) to get qq zzqqkk : ttqq kk qq kk + : qq kk+ : zz qq kk +1 ; qqqq ; zzqq αα. 0: Take aa = tt qq bb = qq pp = qq and αα mm = ( 1) mm qq mmmm mm (qq 3 ; qq 3 ) mm (tt; qq 3 ) mm (zz qq ; qq ) mm zz mm ( zz qq ;qq) mm ( 1 ttqq 4kk 1 )(1 qq kk + )(tt qq qq ; qq) kk qq 3kk +3kk (1 tt qq)(1 qq )(ttqq 3 ; qq 3 ) kk pp kk +5kk ( 1) mm +kk (qq 3 ;qq 3 ) mm+kk (tt; qq 3 ) mm+kk zz (mm +kk ) (zz qq;qq ) mm+kk qq ( mm+kk) αα (mm +kk ) (qq 3 ; qq) mm +kk ( zz qq;qq) mm +kk mm =0

7 Applied Mathematics 01 (5): = ( 1) mm qq mm 3mm +mmmm (tt; qq) mm (zz qq ; qq ) mm zz mm ( zz. qq ; qq) mm mm =0 The right hand side of (6.5) is equal to (tt;qq) Φ(tt αα zz; qq) The left hand side of (6.5) is equal to ( 1) kk (1 ttqq 4kk 1 )(1 qq kk + )(tt; qq) kk 1 (zz qq ; qq ) kk qq kk 3kk +kkkk zz kk (1 qq kk + )( zz qq; qq) kk kk=0 ( 1) mm (qq 3kk+3 ; qq 3 ) mm (tt qq 3kk ; qq 3 ) mm (zz qq kk 1 ;qq ) mm qq mmmm mm zz mm (qq kk+3 ; qq) mm ( zz qq kk 1 ;qq ) mm ( zz qq kk ; qq ) mm mm =0 = ( 1) kk (1 ttqq 4kk 1 )(1 qq kk+ )(tt; qq) kk 1 (zz qq; qq ) kk qq kk 3kk +kkkk zz kk (1 qq kk+ )( zz qq; qq) kk qq 0:zz qq kk 1 0: ttqq 3kk qq 3kk +3 : qq kk +3 : zz qq kk 1 zz qq kk : 00: ; qq qq qq 3 ; zz qq αα which proves Theorem 4(a)(i). Proof of (ii) Take aa = tt qq bb = qq pp = qq and αα mm = ( 1) mm qq mmmm (qq 3 ;qq 3 ) mm (tt; qq 3 ) mm (zz qq ; qq mm zz (qq 3 ; qq) mm ( zz in (6.3) qq ; qq Proof of (iii) Take aa = tt qq bb = qq pp = qq and αα mm = qqmmmm (qq ;qq ) mm (tt qq ; qq ) mm ( zz;qq) mm zz mm (zz qq ;qq in (6.) ) mm+1 Proof of (i v) Take aa = tt qq bb = qq pp = qq and αα mm = qqmm (αα ) (qq 3 ;qq 3 ) mm (tt; qq 3 ) mm zz mm (νν zz;qq) mm (νν 4 in (6.3) zz; qq) mm Proof of (v) Take aa = tt qq bb = qq pp = qq and αα mm = qqmmmm (qq ;qq ) mm (tt; qq ) mm ( zz qq ; qq zz mm+1 (qq ; qq) mm (zz qq ; qq in (6.) Theorem 4(b) The expansions for generalized functions of mock theta functions of third order. (i) ff(tt αα ββ zz; qq) = 1 ( 1 ttqq 4kk 1)(1 qq kk +)(tt; qq) kk 1 qq kk 4kk +kkkk αα kk zz kk (tt) (1 qq kk+ )( zz;qq) kk ( zzzz qq; qq) kk qq 000: ttqq 3kk qq 3kk+3 : qq kk+3 zzqq kk zzzzqq kk 1 : 00: ;qq qq3 ;ααzz qq ββ 3. (ii)φφ(tt αα ββ zz; qq) = 1 ( 1 ttqq 4kk 1)(1 qq kk +)(tt; qq) kk 1 qq kk 3kk +kkkk zz kk (tt) (1 qq kk + )( ααzz qq ; qq ) kk qq 0: 0: ttqq3kk qq 3kk +3 : qq kk +3 : ααzz qq kk 1 : 00: ; qq qq qq 3 ;zz qq ββ. (iii) ψψ(tt αα ββ zz; qq) = 1 ( 1 ttqq 4kk 1)(1 qq kk +)(tt; qq) kk 1 qq kk kk+kkkk zz kk +1 (tt) (1 qq kk + )(ααzz qq ; qq ) kk +1 qq 0: 0:ttqq3kk qq 3kk +3 : qq kk+3 : ααzz qq kk : 00: ; qq qq qq 3 ;zz qq ββ. (iv)νν(tt αα ββ zz; qq) = 1 ( 1 ttqq 4kk 1)(1 qq kk+)(tt; qq) kk 1 qq kk kk +kkkk zz kk (tt) (1 qq kk + )( αα zz qq 3 ;qq ) kk +1 qq 0: 0: ttqq 3kk qq 3kk +3 : qq kk +3 : αα zz qq kk 1 : 00: ; qq qq qq 3 ;zz qq ββ 1. (v)χχ(tt ββ zz; qq) = 1 ( 1 ttqq 4kk 1)(1 qq kk+1)(tt; qq) kk 1 qq kk 3kk +kkkk zz kk (tt) (1 qq kk+1 )(νννν; qq) kk ( νν zz; qq) kk qq 000: ttqq 3kk +1 qq 3kk +3 : qq kk + ννννqq kk νν zzzz kk : 00: ;qq qq3 ;zz qq ββ 3.

8 164 Sameena Saba: A Study of a Generalization of Ramanujan s Third Order and Sixth Order Mock Theta Functions (vi)ρρ(tt ββ zz; qq) = zz4 ( 1 ttqq 6kk 1)(1 qq 4kk +4)(tt; qq) kk 1 qq kk 3kk +kkkk zz 4kk qq 4 (tt) (1 qq kk +4 )(νν zz qq ; qq ) kk +1 (νν zz qq; qq ) kk +1 qq 0: 00: ttqq 5kk qq 5kk +5 : qq kk +5 :νν zz qq kk+1 νν zz qq kk+1 : 00: ;qq qq qq 5 ; zz 4 qq ββ 1. (vii) ωω(tt αα ββ zz;qq) = zz4 ( 1 ttqq 6kk 1)(1 qq 4kk +4)(tt; qq) kk 1 qq kk 5kk +kkkk αα kk zz 4kk qq 4 (tt) (1 qq kk +4 )(zz qq;qq ) kk +1 (αα zz qq 3 ;qq ) kk +1 qq 0: 00: ttqq 5kk qq 5kk +5 : qq kk +5 :zz qq kk +1 αα zz qq kk 1 : 00: ; qq qq qq 5 ;αα zz 4 qq ββ 3. Proof of (i) Take aa = tt qq bb = qq pp = qq and αα mm = qqmmmm 3mm (qq 3 ;qq 3 ) mm (tt; qq 3 ) mm αα mm zz mm (qq 3 in (6.3) ;qq) mm ( zz; qq) mm ( αzz qq ; qq) mm Proof of (ii) Take aa = tt qq bb = qq pp = qq and αα mm = qqmmmm mm (qq 3 ;qq 3 ) mm (tt; qq 3 ) mm zz mm ( ααzz qq ;qq in (6.3) ) mm Proof of (iii) Take aa = tt qq bb = qq qq mmmm (qq 3 ;qq 3 ) mm (tt; qq 3 ) mm zz mm pp = qq and αα mm = (νννν;qq) mm (ααzz qq ; qq in (6.3) Proof of (i v) Take aa = tt qq bb = qq pp = qq and αα mm = qqmmmm mm (qq 3 ; qq 3 ) mm (tt; qq 3 ) mm zz mm ( αα zz qq 3 ; qq in (6.3) Proof of (v) Take aa = tt qq bb = qq pp = qq and αα mm = qqmmmm 3mm (qq 3 ;qq 3 ) mm (ttqq; qq 3 ) mm zz mm (qq ;qq) mm (νννν; qq) mm ( νν in (6.3) zz; qq) mm Proof of (vi ) Take aa = tt qq bb = qq qq mmmm mm (qq 5 ;qq 5 ) 4 mm (tt ;qq 5 ) mm zz 4mm pp = qq and αα mm = (qq 5 ;qq) mm (νν zz qq ;qq ) mm+1 (νν zz qq; qq in (6.4) Proof of (vii ) Take aa = tt qq bb = qq qq mmmm 3mm (qq 5 ;qq 5 ) 4 mm (tt; qq 5 ) mm αα mm zz 4mm pp = qq and αα mm = (qq 5 ;qq) mm (zz qq; qq (αα zz qq 3 ; qq in (6.4) 7. Conclusions Mock theta functions are mysterious functions. These investigations will be helpful in understanding more about these functions. Being shown that they belong to the class of FF qq -functions more properties can be established and relations between these mock theta functions can also be derived. ACKNOWLEDGEMENTS I am thankful to Dr. Bhaskar Srivastava for his help and guidance. REFERENCES [1] G.E. Andrews On basic hypergeometric mock theta functions and partitions (1) Quart. J. Math. 17 (1966) [] G.E. Andrews and D. Hickerson Ramanujan s Lost Notebook-VII: The sixth order mock theta functions Adv. in Math. 89 (1991) [3] G.E. Andrews and B.C. Berndt Ramanujan s Lost Notebook Part I Springer New York (5). [4] G.E. Andrews and B.C. Berndt Ramanujan s Lost Notebook Part II Springer New York (9). [5] Y.S. Choi The Basic Bilateral Hypergeometric Series and the Mock Theta Functions Ramanujan J. 4 (0) [6] Y.S. Choi Tenth order mock theta functions in Ramanujan s lost notebook IV Trans. Amer. Math.Soc. 354 () [7] G. Gasper and M. Rahman Basic Hypergeometric Series Cambridge University Press Cambridge (1990). [8] E.D. Rainville Special Function Chelsea Publishing Company Bronx New York (1960). [9] S. Ramanujan Collected Papers Cambridge University Press 197 reprinted Chelsea New York (196). [10] C.D. Savage and A.J. Yee Euler s partition theorem and the combinatorics of ll-sequences J. Combin. Thy. Ser. A 5 (8) [] Bhaskar Srivastava Ramanujan s fifth order and tenth order mock theta functions- A generalization (Communicated).

9 Applied Mathematics 01 (5): [1] B. Srivastava A Comprehensive study of second order mock theta functions Bull. Korean Math. Soc. 4 No. 4 (5) [13] B. Srivastava A Study of FF qq -functions connected with Ramanujan tenth order Mock theta functions Math. J. Okayam Univ. 46 (4) [14] Bhaskar Srivastava On a generalization of Ramanujan s seventh order mock theta functions (Communicated). [15] G.N. Watson The final problem: an account of the mock theta functions J. London Math. Soc. (1936)