Skip to main content Accessibility help
×
Home
Hostname: page-component-66d7dfc8f5-cqh2z Total loading time: 0.802 Render date: 2023-02-08T08:51:22.261Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Two general series identities involving modified Bessel functions and a class of arithmetical functions

Published online by Cambridge University Press:  10 October 2022

Bruce C. Berndt
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL61801, USA e-mail: berndt@illinois.edu
Atul Dixit
Affiliation:
Department of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, Gujarat382355, India e-mail: adixit@iitgn.ac.in
Rajat Gupta*
Affiliation:
Department of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, Gujarat382355, India. Current address: Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei106319, Taiwan
Alexandru Zaharescu
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL61801, USA, and Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, BucharestRO-70700, Romania e-mail: zaharesc@illinois.edu

Abstract

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty $, generated by Dirichlet series

$$ \begin{align*}\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},\end{align*} $$

satisfying a familiar functional equation involving the gamma function $\Gamma (s)$. Two general identities are established. The first involves the modified Bessel function $K_{\mu }(z)$, and can be thought of as a ‘modular’ or ‘theta’ relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_{\mu }(z)$, the Bessel functions of imaginary argument $I_{\mu }(z)$, and ordinary hypergeometric functions ${_2F_1}(a,b;c;z)$. Although certain special cases appear in the literature, the general identities are new. The arithmetical functions appearing in the identities include Ramanujan’s arithmetical function $\tau (n)$, the number of representations of n as a sum of k squares $r_k(n)$, and primitive Dirichlet characters $\chi (n)$.

Type
Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The first and second authors sincerely thank the MHRD SPARC project SPARC/2018-2019/P567/SL for their financial support. The first author is also supported by a grant from the Simons Foundation. The third author is a postdoctoral fellow at IIT Gandhinagar supported, in part, by the grant CRG/2020/002367.

References

Andrews, G. E., Askey, R. A., and Roy, R., Special functions, Cambridge University Press, Cambridge, 1999.CrossRefGoogle Scholar
Andrews, G. E. and Berndt, B. C., Ramanujan’s lost notebook, part IV, Springer, New York, 2013.CrossRefGoogle Scholar
Berndt, B. C., Identities involving the coefficients of a class of Dirichlet series. III. Trans. Amer. Math. Soc. 146(1969), 323348.CrossRefGoogle Scholar
Berndt, B. C., Identities involving the coefficients of a class of Dirichlet series. V. Trans. Amer. Math. Soc. 160(1971), 139156.CrossRefGoogle Scholar
Berndt, B. C., Dixit, A., Gupta, R., and Zaharescu, A., A class of identities associated with Dirichlet series satisfying Hecke’s functional equation. Proc. Amer. Math. Soc. 150(2022), no. 11, 47854799.Google Scholar
Berndt, B. C., Dixit, A., Kim, S., and Zaharescu, A., Sums of squares and products of Bessel functions. Adv. Math. 338(2018), 305338.CrossRefGoogle Scholar
Berndt, B. C., Lee, Y., and Sohn, J., Koshliakov’s formula and Guinand’s formula in Ramanujan’s lost notebook. In: Alladi, K. (ed.), Surveys in number theory, Springer, New York, 2008, pp. 2142.Google Scholar
Bochner, S., Some properties of modular relations. Ann. Math. 53(1951), 332363.CrossRefGoogle Scholar
Chandrasekharan, K. and Narasimhan, R., Hecke’s functional equation and arithmetical identities. Ann. Math. 4(1961), 123.CrossRefGoogle Scholar
Cohen, H., A course in computational algebraic number theory, Springer, Berlin, 1993.CrossRefGoogle Scholar
Davenport, H., Multiplicative number theory, 3rd. ed., Springer, New York, 2000.Google Scholar
Dixon, A. L. and Ferrar, W. L., Some summations over the lattice points of a circle (I). Quart. J. Math. 5(1934), 4863.CrossRefGoogle Scholar
Edwards, H. M., Riemann’s zeta function, Academic Press, New York, 1974.Google Scholar
Fock, V. A., Zur Berechnung des elektromagnetischen Wechselstromfeldes bei ebener Begrenzung. Ann. Phys. 17(1933), no. 5, 401420.CrossRefGoogle Scholar
Fock, V. A. and Bursian, V., Electromagnetic field of alternating current in a circuit with two groundings, J. Russian Phys. Chem. Soc. 58(2) (1926), 355363 (in Russian).Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series and products, 5th ed., Academic Press, San Diego, 1994.Google Scholar
Guinand, A. P., Some rapidly convergent series for the Riemann $\xi$-function. Quart. J. Math. (Oxford) 6(1955), 156160.CrossRefGoogle Scholar
Hardy, G. H., On the expression of a number as the sum of two squares, Quart. J. Pure Appl. Math. 46 (1915), 263283.Google Scholar
Koshliakov, N. S., On some summation formulae connected with the theory of numbers. II. C. R. Acad. Sci. URSS 1(1934), 553556 (in Russian).Google Scholar
Koshliakov, N. S., On a certain definite integral connected with the cylindric function ${\mathrm{J}}_{\unicode{x3bc}}(\mathrm{x})$. C. R. Acad. Sci. URSS 2(1934), 145147.Google Scholar
Maass, H., Über eine neue art von nichtanalytischen automorphen Funktionen and die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 121(1949), 141183.CrossRefGoogle Scholar
Oberhettinger, F. and Soni, K., On some relations which are equivalent to functional equations involving the Riemann zeta function. Math. Z. 127(1972), 1734.CrossRefGoogle Scholar
Prudnikov, A. P., Brychkov, Y. A., and Marichev, O. I., Integrals and series, vol. 3, Gordon and Breach, New York, 2003.Google Scholar
Ramanujan, S., The lost notebook and other unpublished papers, Narosa, New Delhi, 1988.Google Scholar
Watson, G. N., Some self-reciprocal functions. Quart. J. Math. (Oxford) 2(1931), 298309.Google Scholar
Watson, G. N., Theory of Bessel functions, 2nd ed., Cambridge University Press, London, 1966.Google Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Two general series identities involving modified Bessel functions and a class of arithmetical functions
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Two general series identities involving modified Bessel functions and a class of arithmetical functions
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Two general series identities involving modified Bessel functions and a class of arithmetical functions
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *