# expansions in series of Ferrers functions

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##### 3: 14.18 Sums
###### §14.18 Sums
For expansions of arbitrary functions in series of Legendre polynomials see §18.18(i), and for expansions of arbitrary functions in series of associated Legendre functions see Schäfke (1961b). … The formulas are also valid with the Ferrers functions as in §14.3(i) with $\mu=0$. …
##### 4: 14.32 Methods of Computation
###### §14.32 Methods of Computation
Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. In particular, for small or moderate values of the parameters $\mu$ and $\nu$ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). …
• Application of the uniform asymptotic expansions for large values of the parameters given in §§14.15 and 14.20(vii)14.20(ix).

• ##### 5: 14.15 Uniform Asymptotic Approximations
###### §14.15 Uniform Asymptotic Approximations
In other words, the convergent hypergeometric series expansions of $\mathsf{P}^{-\mu}_{\nu}\left(\pm x\right)$ are also generalized (and uniform) asymptotic expansions as $\mu\to\infty$, with scale $\ifrac{1}{\Gamma\left(j+1+\mu\right)}$, $j=0,1,2,\dots$; compare §2.1(v). … For asymptotic expansions and explicit error bounds, see Dunster (2003b). … For convergent series expansions see Dunster (2004). …
##### 6: 30.4 Functions of the First Kind
###### §30.4(i) Definitions
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(iii) Power-SeriesExpansion
The expansion (30.4.7) converges in the norm of $L^{2}(-1,1)$, that is, …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 $-1\leq x\leq 1$. …
##### 7: 14.13 Trigonometric Expansions
###### §14.13 Trigonometric Expansions
These Fourier series converge absolutely when $\Re\mu<0$. … In particular, …
14.13.4 $\mathsf{Q}_{n}\left(\cos\theta\right)=\frac{2^{2n+1}(n!)^{2}}{(2n+1)!}\*\sum_{% k=0}^{\infty}\frac{1\cdot 3\cdots(2k-1)}{k!}\*\frac{(n+1)(n+2)\cdots(n+k)}{(2n% +3)(2n+5)\cdots(2n+2k+1)}\*\cos\left((n+2k+1)\theta\right),$
For other trigonometric expansions see Erdélyi et al. (1953a, pp. 146–147).
##### 8: Bibliography V
• H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.
• R. S. Varma (1941) An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
• A. N. Vavreck and W. Thompson (1984) Some novel infinite series of spherical Bessel functions. Quart. Appl. Math. 42 (3), pp. 321–324.
• H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.
• H. Volkmer (2021) Fourier series representation of Ferrers function ${\sf P}$ .
• ##### 9: Bibliography C
• B. C. Carlson (2008) Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.
• T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.
• 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.
• J. P. Coleman and A. J. Monaghan (1983) Chebyshev expansions for the Bessel function $J_{n}(z)$ in the complex plane. Math. Comp. 40 (161), pp. 343–366.
• ##### 10: 30.16 Methods of Computation
For small $|\gamma^{2}|$ we can use the power-series expansion (30.3.8). …If $|\gamma^{2}|$ is large we can use the asymptotic expansions in §30.9. … If $|\gamma^{2}|$ is large, then we can use the asymptotic expansions referred to in §30.9 to approximate $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$. … A fourth method, based on the expansion (30.8.1), is as follows. … …