scholarly journals An exact integral-to-sum relation for products of Bessel functions

2021 ◽  
Vol 477 (2253) ◽  
pp. 20210376
Author(s):  
Oliver H.E. Philcox ◽  
Zachary Slepian
Keyword(s):  
Bessel Functions ◽  
Summation Formula ◽  
Poisson Summation Formula ◽  
Infinite Integral ◽  
Infinite Sum ◽  
Poisson Summation

A useful identity relating the infinite sum of two Bessel functions to their infinite integral was discovered in Dominici et al. (Dominici et al. 2012 Proc. R. Soc. A 468 , 2667–2681). Here, we extend this result to products of N Bessel functions, and show it can be straightforwardly proven using the Abel-Plana theorem, or the Poisson summation formula. For N  = 2, the proof is much simpler than that of Dominici et al. and significantly enlarges the range of validity.

scholarly journals Some Finite Analogues of the Poisson Summation Formula

1961 ◽  
Vol 12 (3) ◽  
pp. 133-138 ◽  
Author(s):  
L. Carlitz
Keyword(s):  
Summation Formula ◽  
Poisson Summation Formula ◽  
Euler’S Constant ◽  
Image Position ◽  
Positive Integers ◽  
Euler's Constant ◽  
Poisson Summation

1. Guinand (2) has obtained finite identities of the typewhere m, n, N are positive integers and eitherorwhere γ is Euler's constant and the notation ∑′ indicates that when x is integral the term r = x is multiplied by ½. Clearly there is no loss of generality in taking N = 1 in (1.1).

scholarly journals A strong version of Poisson summation

1997 ◽  
Vol 20 (1) ◽  
pp. 81-92 ◽  
Author(s):  
Nelson Petulante
Keyword(s):  
Summation Formula ◽  
Poisson Summation Formula ◽  
Strong Version ◽  
Poisson Summation

We establish a generalized version of the classical Poisson summation formula. This formula incorporates a special feature called “compression”, whereby, at the same time that the formula equates a series to its Fourier dual, the compressive feature serves to enable both sides of the equation to converge.

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