Recurrences Related to the Bessel Function

Abstract

For k = 0, 1, 2, … let J_k(z) be the Bessel function of the first kind. Put

$$ {(z/2)^k}/{J_k}(z) = \sum\limits_{n = 0}^\infty {{u_n}(k)\frac{{{{(z/2)}^{2n}}}}{{n!(n + k)!}}} $$

((1.1))

; then if follows that u₀(k) =(k!)², and for n > 0

$$ \sum\limits_{r = 0}^n {{{( - 1)}^r}\left( \begin{array}{l} n + k \\ r + k \\ \end{array} \right)} \;\left( \begin{array}{l} n + k \\ r \\ \end{array} \right){u_r}(k) = 0 $$

.