Transformation formulas associated with integrals involving the Riemann Ξ-function and rank-crank type PDEs in partition theory

Dixit, Atul

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Description

Title

Transformation formulas associated with integrals involving the Riemann Ξ-function and rank-crank type PDEs in partition theory

Author(s)

Dixit, Atul

Issue Date

2012-09-18T21:12:47Z

Director of Research (if dissertation) or Advisor (if thesis)

Berndt, Bruce C.

Doctoral Committee Chair(s)

Kevin Ford

Committee Member(s)

• Berndt, Bruce C.
• Stolarsky, Kenneth B.
• Hildebrand, A.J.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Ramanujan
• Riemann zeta function
• Hurwitz zeta function
• Bessel function
• Koshliakov
• Hardy
• Ferrar
• Rank-Crank partial differential equation (Rank-Crank PDE)
• Partition
• $q$-series
• Eisenstein series
• Appell function

Abstract

There are two parts to this thesis. The first part deals with obtaining ``modular''-type transformation formulas involving various special functions such as the digamma function, Hurwitz zeta function and modified Bessel functions, through integrals involving the Riemann $\Xi$-function. Our motivation is a beautiful formula involving the digamma and the Riemann $\Xi$-functions found on page 220 in the volume ``Ramanujan's lost notebook and other unpublished papers''. In Chapter 2, we give two different proofs of this result, the second one being implicitly sought by A.P.~Guinand while rediscovering this result. However, Guinand only partially rediscovered it in the sense that he did not see the connection with the integral involving the Riemann $\Xi$-function. A simple and cute, yet very useful, trick based upon the invariance of this integral under a certain map is used to provide the proof of the complete identity of Ramanujan. From Chapter 3 onwards, we consider general integrals involving the Riemann $\Xi$-function and evaluate these using Mellin transforms and residue calculus. Then we use the above-mentioned trick to obtain modular transformation formulas. This provides a very convenient way of generating these formulas without using Poisson summation or Fourier transforms. Chapter 3 is devoted to transformation formulas of the type $F(\alpha)=F(\beta)$, where $\alpha\beta=1$. Here, we unify many well-known formulas in the literature such as Koshliakov's formula, Hardy's formula, Ferrar's formula and, of course, Ramanujan's above transformation formula, in the sense that all of them can be generated in similar ways starting from the general integral involving the Riemann $\Xi$-function. This also leads us to extended versions of Koshliakov's formula and Ferrar's formula. In Chapter 4, we study one-parameter generalizations of the extended versions of Koshliakov's and Ramanujan's formulas from Chapter 3, i.e., we obtain formulas of the type $F(z,\alpha)=F(z,\beta)$, where $\alpha\beta=1$. In this case, we need to consider integrals involving a product of two Riemann $\Xi$-functions, the prototypical example of which was given by Ramanujan in his paper ``New expressions for Riemann's functions $\xi(s)$ and $\Xi(t)$''. Here we obtain a new proof as well as an extended version of the Ramanujan-Guinand formula. Also, Ramanujan's formula is generalized to a new one involving the Hurwitz zeta function. New analogues of this generalization are also obtained. Furthermore, we end this chapter by discussing proofs of two of the three transformation formulas involving the Hurwitz zeta function through special functions and without using contour integration and Mellin transforms. The proofs require certain identitites of Ramanujan from the above-referred paper. However, two identities in the paper are incorrect. Here, we derive correct versions of the same. Chapter 5 is devoted to analogues of some of the results from Chapters 3 and 4 for primitive Dirichlet characters. These are of the form $F(z, \alpha,\chi)=F(-z, \beta,\overline{\chi})=F(-z,\alpha,\overline{\chi})=F(z,\beta,\chi)$, where $\alpha\beta=1$. Character analogues of the Hurwitz zeta function and digamma function make their appearance here. Chapter 6 contains material from joint work with Bruce C.~Berndt and Jaebum Sohn and involves a different character analogue of the Ramanujan-Guinand formula. We also give character analogues of the results found on pages 253--254 in Ramanujan's lost notebook. However, here we give results for only even primitive characters. The ones for even as well as odd primitive characters can be found in \cite{bds}. A new topic which fills the remainder of this thesis is discussed in Chapter 7. This is a topic in Partition Theory and it is concerned with obtaining Rank-Crank type PDE's through certain bilateral basic hypergeometric series identities due to M.~Jackson and S.H.~Chan. The motivation comes from the fact that the original Rank-Crank PDE due to A.O.L.~Atkin and F.G.~Garvan was obtained through an identity of Atkin and H.P.F.~Swinnerton-Dyer, which is a special case of Chan's identity. We give proof of only the fourth order PDE of Garvan and not of the general PDE. This contains part of the material from joint work with Chan and Garvan \cite{dg}. The proof given here is different from the one given there.

Graduation Semester

2012-08

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Copyright 2012 Atul Dixit

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