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ON AN INTEGRAL OF $\boldsymbol {J}$-BESSEL FUNCTIONS AND ITS APPLICATION TO MAHLER MEASURE

Part of: Stochastic processes

Published online by Cambridge University Press:  09 July 2021

GEORGE ANTON Open the ORCID record for GEORGE ANTON [Opens in a new window] ,
JESSEN A. MALATHU Open the ORCID record for JESSEN A. MALATHU [Opens in a new window] ,
SHELBY STINSON Open the ORCID record for SHELBY STINSON [Opens in a new window]  and
J. S. Friedman
GEORGE ANTON
Affiliation:
Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, NY10031, USA e-mail: ganton000@citymail.cuny.edu, SendittoGeorgeAnton@gmail.com
JESSEN A. MALATHU*
Affiliation:
Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, NY10031, USA
SHELBY STINSON
Affiliation:
Department of Mathematics, Fordham University, 441 E. Fordham Road, Bronx, NY 10458, USA e-mail: sstinson1@fordham.edu, shstinson96@gmail.com
J. S. Friedman*
Affiliation:
Department of Mathematics and Science, United States Merchant Marine Academy, 300 Steamboat Road, Kings Point, NY 11024, USA e-mail: FriedmanJ@usmma.edu, joshua@math.sunysb.edu, CrownEagle@gmail.com
*
e-mail: jmalath000@citymail.cuny.edu, jmalathu@hotmail.com
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Abstract

Cogdell et al. [‘Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space’, Trans. Amer. Math. Soc. (2021), to appear] developed infinite series representations for the logarithmic Mahler measure of a complex linear form with four or more variables. We establish the case of three variables by bounding an integral with integrand involving the random walk probability density $a\int _0^\infty tJ_0(at) \prod _{m=0}^2 J_0(r_m t)\,dt$ , where $J_0$ is the order-zero Bessel function of the first kind and a and $r_m$ are positive real numbers. To facilitate our proof we develop an alternative description of the integral’s asymptotic behaviour at its known points of divergence. As a computational aid for numerical experiments, an algorithm to calculate these series is presented in the appendix.

Keywords

Mahler measurerandom walkBessel function

MSC classification

Secondary: 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$ 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 2 , April 2022 , pp. 223 - 235
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

The authors acknowledge the support of NSF grant DMS-1820731.

†

The views expressed in this article are the author’s own and not those of the United States Merchant Marine Academy, the Maritime Administration, the Department of Transportation or the United States government.

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