# §30.1 Special Notation

(For other notation see Notation for the Special Functions.)

$x$ real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, $-1. real parameter (positive, zero, or negative). order, a nonnegative integer. degree, an integer $n=m,m+1,m+2,\dots$. integer. arbitrary small positive constant.

The main functions treated in this chapter are the eigenvalues $\lambda^{m}_{n}\left(\gamma^{2}\right)$ and the spheroidal wave functions $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$, $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)$, $\mathit{Ps}^{m}_{n}\left(z,\gamma^{2}\right)$, $\mathit{Qs}^{m}_{n}\left(z,\gamma^{2}\right)$, and $S^{m(j)}_{n}\left(z,\gamma\right)$, $j=1,2,3,4$. These notations are similar to those used in Arscott (1964b) and Erdélyi et al. (1955). Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathsf{Ps}$, $\mathsf{Qs}$, $\mathit{Ps}$, $\mathit{Qs}$, respectively.

## Other Notations

Flammer (1957) and Abramowitz and Stegun (1964) use $\lambda_{mn}(\gamma)$ for $\lambda^{m}_{n}\left(\gamma^{2}\right)+\gamma^{2}$, $R_{mn}^{(j)}(\gamma,z)$ for $S^{m(j)}_{n}\left(z,\gamma\right)$, and

 30.1.1 $\displaystyle S^{(1)}_{mn}(\gamma,x)$ $\displaystyle=d_{mn}(\gamma)\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),$ $\displaystyle S^{(2)}_{mn}(\gamma,x)$ $\displaystyle=d_{mn}(\gamma)\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right),$

where $d_{mn}(\gamma)$ is a normalization constant determined by

 30.1.2 $\displaystyle S^{(1)}_{mn}(\gamma,0)$ $\displaystyle=(-1)^{m}\mathsf{P}^{m}_{n}\left(0\right),$ $n-m$ even, $\displaystyle\left.\frac{\mathrm{d}}{\mathrm{d}x}S^{(1)}_{mn}(\gamma,x)\right|% _{x=0}$ $\displaystyle=(-1)^{m}\left.\frac{\mathrm{d}}{\mathrm{d}x}\mathsf{P}^{m}_{n}% \left(x\right)\right|_{x=0},$ $n-m$ odd.

For older notations see Abramowitz and Stegun (1964, §21.11) and Flammer (1957, pp. 14,15).