# Analogues of the general theta transformation formula

@article{Dixit2013AnaloguesOT, title={Analogues of the general theta transformation formula}, author={Atul Dixit}, journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, year={2013}, volume={143}, pages={371 - 399} }

A new class of integrals involving the confluent hypergeometric function 1F1(a;c;z) and the Riemann Ξ-function is considered. It generalizes a class containing some integrals of Ramanujan, Hardy and Ferrar and gives, as by-products, transformation formulae of the form F(z, α) = F(iz, β), where αβ = 1. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulae of Ferrar and Hardy. A one-variable generalization of a… Expand

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#### References

SHOWING 1-10 OF 45 REFERENCES

Character analogues of Ramanujan-type integrals involving the Riemann Ξ-function

- Mathematics
- 2011

A new class of integrals involving the product of4-functions associated with primitive Dirichlet characters is considered. These integrals give rise to transformation formulas of the type F.z;;/ D F.… Expand

Series transformations and integrals involving the Riemann Ξ-function

- Mathematics
- 2010

The transformation formulas of Ramanujan, Hardy, Koshliakov and Ferrar are unified, in the sense that all these formulas come from the same source, namely, a general formula involving an integral of… Expand

Analogues of a transformation formula of Ramanujan

- Mathematics
- 2009

We derive two new analogues of a transformation formula of Ramanujan involving the Gamma and Riemann zeta functions present in the Lost Notebook. Both involve infinite series consisting of Hurwitz… Expand

Transformation formulas associated with integrals involving the Riemann Ξ-function

- Mathematics
- 2011

Using residue calculus and the theory of Mellin transforms, we evaluate integrals of a certain type involving the Riemann Ξ-function, which give transformation formulas of the form F(z, α) = F(z, β),… Expand

A proof of the general theta transformation formula

- Mathematics
- 2010

The most familiar proof of Theorem 1.1, especially in the form (1.2), is via Poisson’s summation formula. That proof is briefly sketched in [2, p. 253]. It should be emphasized that only in special… Expand

A transformation formula involving the gamma and riemann zeta functions in Ramanujan's lost notebook

- Mathematics
- 2010

Two proofs are given for a series transformation formula involving the logarithmic derivative of the Gamma function found in Ramanujan’s lost notebook. The transformation formula is connected with a… Expand

Certain summation and transformation formulas for generalized hypergeometric series

- Computer Science, Mathematics
- J. Comput. Appl. Math.
- 2009

A reduction formula for a certain Kampe de Feriet function is deduced that provides a Kummer-type transformation formula for the generalized hypergeometric function "pF"p(x) and reduces to Minton's summation theorem. Expand

A Kummer-type transformation for a 2 F 2 hypergeometric function

- Mathematics
- 2005

We obtain a Kummer-type transformation for the 2 F 2 ( x ) hypergeometric function with general parameters in the form of a sum of 2 F 2 ( - x ) functions. This result is specialised to the case… Expand

The Theory of the Riemann Zeta-Function

- Mathematics
- 1987

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects… Expand

Asymptotics and Mellin-Barnes Integrals

- Mathematics
- 2001

Asymptotics and Mellin-Barnes Integrals provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typically… Expand