A new proof of Sonine's formula

Author:
Krzysztof Stempak

Journal:
Proc. Amer. Math. Soc. **104** (1988), 453-457

MSC:
Primary 33A40

MathSciNet review:
962812

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Abstract: We give a new proof of an old integral formula due to Sonine in which the Bessel functions are involved. By considering the Banach algebra of radial functions on , we observe that Sonine's formula is valid for all positive integers. Next, a complex function theory argument is applied to obtain the validity of the formula for all complex parameters with Re .

**[1]**Billie Chandler Carlson,*Special functions of applied mathematics*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. MR**0590943****[2]**Douglas L. Guy,*Hankel multiplier transformations and weighted 𝑝-norms*, Trans. Amer. Math. Soc.**95**(1960), 137–189. MR**0120506**, 10.1090/S0002-9947-1960-0120506-1**[3]**E. C. Titchmarsh,*The theory of functions*, Oxford Univ. Press, London, 1939.**[4]**G. N. Watson,*A treatise on the theory of Bessel functions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR**1349110**

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0962812-X

Keywords:
Bessel functions,
radial functions

Article copyright:
© Copyright 1988
American Mathematical Society