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Summing Sneddon-Bessel series explicitly.
- Source :
-
Mathematical Methods in the Applied Sciences . 5/15/2024, Vol. 47 Issue 7, p6590-6606. 17p. - Publication Year :
- 2024
-
Abstract
- We sum in a closed form the Sneddon-Bessel series ∑∞ m=1 Jα(xjm,v)Jβ(yjm,v) /j m,v2n+α+β-2v+2 Jv+1(jm,v)², where 0 < x, 0 < y, x + y < 2, n is an integer, α, β, v ε C\{-1,-2, ... } with 2 Re v < 2n + 1 + Re α + Re β and {jm,v}m≥0 are the zeros of the Bessel function Jv of order v. In most cases, the explicit expressions for these sums involve hypergeometric functions pFq. As an application, we prove some extensions of the Kneser-Sommerfeld expansion. For instance, we show that ∑∞ m=1 j v-β m,v Jv (xjm,v)Jβ (yjm,v) (j²m,v-z²)Jv+1(jm,v)² = πJβ (yz)/4zβ-v Jv (z) (Yv (z)Jv (xz) - Jv (z)Yv (xz)), if Re v < Re β + 1 and 0 < y ≤ x, x + y < 2 (here, Yv denotes the Bessel function of the second kind), which becomes the Kneser-Sommerfeld expansion when β = v. [ABSTRACT FROM AUTHOR]
- Subjects :
- *HYPERGEOMETRIC functions
*BESSEL functions
*HYPERGEOMETRIC series
*INTEGERS
Subjects
Details
- Language :
- English
- ISSN :
- 01704214
- Volume :
- 47
- Issue :
- 7
- Database :
- Academic Search Index
- Journal :
- Mathematical Methods in the Applied Sciences
- Publication Type :
- Academic Journal
- Accession number :
- 177253476
- Full Text :
- https://doi.org/10.1002/mma.9939