# Ferrers functions

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## 31—40 of 47 matching pages

##### 31: 30.4 Functions of the First Kind
If $\gamma=0$, $\mathsf{Ps}^{m}_{n}\left(x,0\right)$ reduces to the Ferrers function $\mathsf{P}^{m}_{n}\left(x\right)$:
30.4.2 $\mathsf{Ps}^{m}_{n}\left(x,0\right)=\mathsf{P}^{m}_{n}\left(x\right);$
It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for $-1\leq x\leq 1$. …
##### 32: 30.1 Special Notation
(For other notation see Notation for the Special Functions.) … 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 $S^{m(j)}_{n}\left(z,\gamma\right)$, $j=1,2,3,4$. …Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathsf{Ps}$, $\mathsf{Qs}$, $\mathit{Ps}$, $\mathit{Qs}$, respectively.
##### 33: Bibliography V
• H. Volkmer (2021) Fourier series representation of Ferrers function ${\sf P}$ .
• ##### 34: 10.43 Integrals
10.43.22 $\int_{0}^{\infty}t^{\mu-1}e^{-at}K_{\nu}\left(t\right)\mathrm{d}t=\begin{cases% }\left(\frac{1}{2}\pi\right)^{\frac{1}{2}}\Gamma\left(\mu-\nu\right)\Gamma% \left(\mu+\nu\right)(1-a^{2})^{-\frac{1}{2}\mu+\frac{1}{4}}\mathsf{P}^{-\mu+% \frac{1}{2}}_{\nu-\frac{1}{2}}\left(a\right),&-1
For the Ferrers function $\mathsf{P}$ and the associated Legendre function $P$, see §§14.3(i) and 14.21(i). …
##### 35: 30.16 Methods of Computation
30.16.9 $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)=\lim_{d\to\infty}\sum_{j=1}^{d}(-% 1)^{j-p}e_{j,d}\mathsf{P}^{m}_{n+2(j-p)}\left(x\right).$
##### 36: 18.3 Definitions
Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions14.7(i)). …
##### 37: Mathematical Introduction
Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). …
##### 38: Bibliography C
• H. S. Cohl and R. S. Costas-Santos (2020) Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).
• H. S. Cohl, J. Park, and H. Volkmer (2021) Gauss hypergeometric representations of the Ferrers function of the second kind. SIGMA Symmetry Integrability Geom. Methods Appl. 17, pp. Paper 053, 33.
• ##### 39: 18.17 Integrals
18.17.7 $\left(P_{n}\left(x\right)\right)^{2}+4\pi^{-2}\left(\mathsf{Q}_{n}\left(x% \right)\right)^{2}=4\pi^{-2}\*\int_{1}^{\infty}Q_{n}\left(x^{2}+(1-x^{2})t% \right)(t^{2}-1)^{-\frac{1}{2}}\mathrm{d}t,$ $-1.
For the Ferrers function $\mathsf{Q}_{n}\left(x\right)$ and Legendre function $Q_{n}\left(x\right)$ see §§14.3(i) and 14.3(ii), with $\mu=0$ and $\nu=n$. …
##### 40: 14.34 Software
A more complete list of available software for computing these functions is found in the Software Index. For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C9).