Lommel function

2020 Mathematics Subject Classification: Primary: 34-XX [MSN][ZBL]

A solution of the non-homogeneous Bessel equation $$x^2y''+xy'+(x^2-\nu^2)y = x^\rho.$$ If $\rho = \nu+2n$, where $n$ is a natural number, then

$$y=(-1)^{n-1}(n-1)!\; 2^{\nu+2n-2} \sum_{k=0}^{n-1} (-1)^k\big(\frac{x}{2}\big)^{\nu+2k} \frac{\def\G{\Gamma}\G(\nu+n)}{k!\G(\nu+k+1)}$$

(see 10.71 in [Wa]). If the numbers $\rho+\nu\ge 0$ and $\rho-\nu\ge 0$ are not integers, then

$$y=2^{\rho -2}\G(\frac{\rho+\nu}{2})\G(\frac{\rho-\nu}{2})\sum_{k=0}^\infty\frac{(-1)^k(x/2)^{\rho+2k}}{\G(k+1+(\rho+\nu)/2)\G(k+1+(\rho-\nu)/2)}$$ (see 10.7 in [Wa]).

If $\rho = \nu-2n$, where $n\ge 0$ is an integer and $\nu$ is not an integer $\le n$, then

$$y=\frac{\G(\nu-n)}{n!\; 2^{-\nu+2n+2}}\Big[2J_\nu(x)\ln\frac{x}{2}-\sum_{k=0}^{n-1}\frac{(n-k-1)!}{\G(\nu-n+k+1)}\big(\frac{x}{2}\big)^{\nu-2n+2k}-\sum_{k=0}^{\infty}\Big(\frac{(-1)^k(x/2)^{\nu+2k}}{k!\;\G(\nu+k+1)}-\frac{\G'(k+1)}{\G(k+1)}-\frac{\G'(\nu+k+1)}{\G(\nu+k+1)}\Big)\Big].$$

Here, for $n=0$ the first sum is taken to be zero, and $J_\nu(x)$ is a Bessel function (cf. Bessel functions). Lommel functions in two variables are also known.

See also Anger function; Weber function; Struve function.

Lommel functions were studied by E. Lommel [Lo].

References

[Ka]	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", 1. Gewöhnliche Differentialgleichungen, Chelsea, reprint (1947)
[Lo]	E. Lommel, "Zur Theorie der Bessel'schen Funktionen IV" Math. Ann., 16 (1880) pp. 183–208 MR1510022 JFM Zbl 12.0773.01 JFM Zbl 12.0398.01
[Wa]	G.N. Watson, "A treatise on the theory of Bessel functions", 1, Cambridge Univ. Press (1952) MR1349110 MR1570252 MR0010746 MR1520278 Zbl 0849.33001 Zbl 0174.36202 Zbl 0063.08184 JFM Zbl 48.0412.02 JFM Zbl 50.0264.01