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## Acta Arithmetica

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## Error functions, Mordell integrals and an integral analogue of a partial theta function

### Tom 177 / 2017

Acta Arithmetica 177 (2017), 1-37 MSC: Primary 11M06, 33B20; Secondary 33C15, 34E05. DOI: 10.4064/aa8207-5-2016 Opublikowany online: 1 December 2016

#### Streszczenie

A new transformation involving the error function $\operatorname{erf} (z)$, the imaginary error function $\operatorname{erfi} (z)$, and an integral analogue of a partial theta function is given along with its character analogues. Another complementary error function transformation is also obtained which when combined with the first explains a transformation in Ramanujan’s Lost Notebook termed by Berndt and Xu as the one for an integral analogue of theta functions. These transformations are used to obtain a variety of exact and approximate evaluations of some non-elementary integrals involving hypergeometric functions. Several asymptotic expansions, including the one for a non-elementary integral involving a product of the Riemann $\Xi$-function of two different arguments, are obtained, which generalize known results due to Berndt and Evans, and Oloa.

#### Autorzy

• Atul DixitDepartment of Mathematics
Tulane University
New Orleans, LA 70118, U.S.A.
Department of Mathematics
Indian Institute of Technology Gandhinagar
Palaj, Gandhinagar 382355, India
e-mail
• Arindam RoyDepartment of Mathematics
University of Illinois
1409 West Green Street
Urbana, IL 61801, U.S.A.
and
Department of Mathematics
Rice University
6100 Main St.
Houston, TX 77005, U.S.A.
e-mail
e-mail
• Alexandru ZaharescuDepartment of Mathematics
University of Illinois
1409 West Green Street
Urbana, IL 61801, U.S.A.
and
Simion Stoilow Institute of Mathematics