Sameena Saba
Karamat Husain Muslim Girls P.G. College Lucknow
Correspondence to: Sameena Saba , Karamat Husain Muslim Girls P.G. College Lucknow.
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Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved.
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 -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
Cite this paper:
Sameena Saba , "A Study of a Generalization of Ramanujan’s Third Order and Sixth Order Mock Theta Functions", Applied Mathematics, Vol. 2 No. 5, 2012, pp. 157-165. doi: 10.5923/j.am.20120205.02.
1. Introduction
S. Ramanujan in his last letter to G.H. Hardy[9, pp 354-355] 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 there is a θ-function such that the differenceis bounded as radially.(1) There is no single θ-function which works for all i.e., for every θ-function there is some root of unity for which difference is unbounded as 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 visiting Trinity 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[2] called them of sixth order. The sixth order mock theta functions of Ramanujan are 2000 Mathematics Subject Classification.33D15Key words and phrases. Basic Hypergeometric series, mock theta functions, q-integral, multibasic expansions. The third order mock theta functions of Ramanujan are We give a generalization of the sixth order and the third order mock theta functions. The generalized sixth order mock theta functions are | (1.1) |
| (1.2) |
| (1.3) |
| (1.4) |
| (1.5) |
| (1.6) |
| (1.7) |
For we have the generalized functions of Choi[4]. For and using induction, these functions satisfy the Ramanujan’s requirement for a mock theta function.The generalized third order mock theta functions are | (1.8) |
| (1.9) |
| (1.10) |
| (1.11) |
| (1.12) |
| (1.13) |
and | (1.14) |
whereFor and we have generalized five third order mock theta functions namely of Andrews[1]. For and the generalized functions and reduce to the third order mock theta functions of Ramanujan and and ρ to the third order mock theta functions of Watson[15].In this study we will show that these generalized functions are-functions. This is done in section 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 give multibasic expansion for these generalized functions.
2. Notations
We shall use the following usual basic hypergeometric notations:
3. The Generalized Functions are -Functions
We show these generalized functions are - functions.Theorem 1The generalized functions and the generalized functions are functions.ProofsWe shall give the proof for only. The proofs for the other functions are similar, hence omitted.Applying the difference operator to we have SoHence is a - 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 2Proof of (i)which proves Theorem 2 (i).Proof of (ii)which proves Theorem 2 (ii).Proof of (iii)Writing for α in , we havewhich proves Theorem 2 (iii).Proof of (iv)Writing for α and then for α in in (1.8), we havewhich proves Theorem 2 (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] asWe 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)ProofWe shall give the detailed proof for The proofs for the other functions are similar, so omitted.Limiting case of q-beta integral[7, p. 19 (1.11.7)] is | (5.1) |
NowReplacing t by and by a, we have | (5.2) |
Butand since ,Hence | (5.3) |
by using (5.3), (5.2) can be written aswhich 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:ProofThe proofs are similar to given above for , 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 | (6.1) |
Corollary 1Letting and in (6.1) we have | (6.2) |
Corollary 2Letting and in (6.1) we have | (6.3) |
Corollary 3Letting and in (6.1) we have | (6.4) |
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 and for other functions we write only the specialized parameters.Theorem 4(a)The expansions for generalized functions of sixth order mock theta functions.Proof of (i)in (6.3) to getThe right hand side of (6.5) is equal toThe left hand side of (6.5) is equal towhich proves Theorem 4(a)(i).Proof of (ii)Proof of (iii)Proof of (iv)Proof of (v)Theorem 4(b)The expansions for generalized functions of mock theta functions of third order.Proof of (i)Proof of (ii)Proof of (iii)Proof of (iv)Proof of (v)Proof of (vi)Proof of (vii)
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 -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
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[2] | G.E. Andrews and D. Hickerson, Ramanujan’s ‘Lost’ Notebook-VII: The sixth order mock theta functions, Adv. in Math. 89 (1991) 60-105. |
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[15] | G.N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc. 11 (1936) 55-80. |