§14.18 Sums

Keywords:: Ferrers functions, associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.18

§14.18(i) Expansion Theorem
§14.18(ii) Addition Theorems
§14.18(iii) Other Sums
§14.18(iv) Compendia

§14.18(i) Expansion Theorem

Keywords:: associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.18.i

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).

§14.18(ii) Addition Theorems

Notes:: See Erdélyi et al. (1953a, pp. 168–169) and Olver (1997b, p. 183). Errors in Erdélyi et al. (1953a) have been corrected. (14.18.2) may be derived from (14.7.16), (14.9.3), and (14.18.1).
Keywords:: Ferrers functions, associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.18.ii

In (14.18.1) and (14.18.2), $\theta _{1}$ , $\theta _{2}$ , and $\theta _{1}+\theta _{2}$ all lie in $[0,\pi)$ , and $\phi$ is real.

14.18.1 $\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{1}\mathop{\cos\/}\nolimits\theta _{2}+\mathop{\sin\/}\nolimits\theta _{1}\mathop{\sin\/}\nolimits\theta _{2}\mathop{\cos\/}\nolimits\phi\right)=\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{1}\right)\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{2}\right)+2\sum _{{m=1}}^{{\infty}}(-1)^{m}\mathop{\mathsf{P}^{{-m}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{1}\right)\mathop{\mathsf{P}^{{m}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{2}\right)\mathop{\cos\/}\nolimits\!\left(m\phi\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $m$ : nonnegative integer, $\nu$ : general degree, $\theta _{1}$ , $\theta _{2}$ : real and $\phi$ : real
Referenced by:: §14.18(ii), §14.18(ii)
Permalink:: http://dlmf.nist.gov/14.18.E1
Encodings:: TeX, pMML, png

14.18.2 $\mathop{\mathsf{P}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{1}\mathop{\cos\/}\nolimits\theta _{2}+\mathop{\sin\/}\nolimits\theta _{1}\mathop{\sin\/}\nolimits\theta _{2}\mathop{\cos\/}\nolimits\phi\right)=\sum _{{m=-n}}^{{n}}(-1)^{m}\mathop{\mathsf{P}^{{-m}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{1}\right)\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{2}\right)\mathop{\cos\/}\nolimits(m\phi).$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $m$ : nonnegative integer, $n$ : nonnegative integer, $\theta _{1}$ , $\theta _{2}$ : real and $\phi$ : real
Referenced by:: §14.18(ii), §14.18(ii)
Permalink:: http://dlmf.nist.gov/14.18.E2
Encodings:: TeX, pMML, png

In (14.18.3), $\theta _{1}$ lies in $(0,\frac{1}{2}\pi)$ , $\theta _{2}$ and $\theta _{1}+\theta _{2}$ both lie in $(0,\pi)$ , $\theta _{1}<\theta _{2}$ , $\phi$ is real, and $\nu\neq-1,-2,-3,\dots$ .

14.18.3 $\mathop{\mathsf{Q}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{1}\mathop{\cos\/}\nolimits\theta _{2}+\mathop{\sin\/}\nolimits\theta _{1}\mathop{\sin\/}\nolimits\theta _{2}\mathop{\cos\/}\nolimits\phi\right)=\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{1}\right)\mathop{\mathsf{Q}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{2}\right)+2\sum _{{m=1}}^{{\infty}}(-1)^{m}\mathop{\mathsf{P}^{{-m}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{1}\right)\mathop{\mathsf{Q}^{{m}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta _{2}\right)\mathop{\cos\/}\nolimits\!\left(m\phi\right).$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $m$ : nonnegative integer, $\nu$ : general degree, $\theta _{1}$ , $\theta _{2}$ : real and $\phi$ : real
Referenced by:: §14.18(ii), Ch.14
Permalink:: http://dlmf.nist.gov/14.18.E3
Encodings:: TeX, pMML, png

In (14.18.4) and (14.18.5), $\xi _{1}$ and $\xi _{2}$ are positive, and $\phi$ is real; also in (14.18.5) $\xi _{1}<\xi _{2}$ and $\nu\neq-1,-2,-3,\dots$ .

14.18.4 $\mathop{P_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi _{1}\mathop{\cosh\/}\nolimits\xi _{2}-\mathop{\sinh\/}\nolimits\xi _{1}\mathop{\sinh\/}\nolimits\xi _{2}\mathop{\cos\/}\nolimits\phi\right)=\mathop{P_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi _{1}\right)\mathop{P_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi _{2}\right)+2\sum _{{m=1}}^{{\infty}}(-1)^{m}\mathop{P^{{-m}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi _{1}\right)\mathop{P^{{m}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi _{2}\right)\mathop{\cos\/}\nolimits\!\left(m\phi\right),$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $m$ : nonnegative integer, $\nu$ : general degree, $\phi$ : real and $\xi _{1}$ , $\xi _{2}$ : positive
Referenced by:: §14.18(ii)
Permalink:: http://dlmf.nist.gov/14.18.E4
Encodings:: TeX, pMML, png

14.18.5 $\mathop{Q_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi _{1}\mathop{\cosh\/}\nolimits\xi _{2}-\mathop{\sinh\/}\nolimits\xi _{1}\mathop{\sinh\/}\nolimits\xi _{2}\mathop{\cos\/}\nolimits\phi\right)=\mathop{P_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi _{1}\right)\mathop{Q_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi _{2}\right)+2\sum _{{m=1}}^{{\infty}}(-1)^{m}\mathop{P^{{-m}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi _{1}\right)\mathop{Q^{{m}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi _{2}\right)\mathop{\cos\/}\nolimits\!\left(m\phi\right).$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $m$ : nonnegative integer, $\nu$ : general degree, $\phi$ : real and $\xi _{1}$ , $\xi _{2}$ : positive
Referenced by:: §14.18(ii)
Permalink:: http://dlmf.nist.gov/14.18.E5
Encodings:: TeX, pMML, png

§14.18(iii) Other Sums

Notes:: See Erdélyi et al. (1953a, pp. 162 and 167). (14.18.8) may be derived from (14.7.17) and (14.18.9).
Permalink:: http://dlmf.nist.gov/14.18.iii

14.18.6 $(x-y)\sum _{{k=0}}^{{n}}(2k+1)\mathop{P_{{k}}\/}\nolimits\!\left(x\right)\mathop{P_{{k}}\/}\nolimits\!\left(y\right)=(n+1)\left(\mathop{P_{{n+1}}\/}\nolimits\!\left(x\right)\mathop{P_{{n}}\/}\nolimits\!\left(y\right)-\mathop{P_{{n}}\/}\nolimits\!\left(x\right)\mathop{P_{{n+1}}\/}\nolimits\!\left(y\right)\right),$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind and $n$ : nonnegative integer
A&S Ref:: 8.9.1
Permalink:: http://dlmf.nist.gov/14.18.E6
Encodings:: TeX, pMML, png

14.18.7 $(x-y)\sum _{{k=0}}^{{n}}(2k+1)\mathop{P_{{k}}\/}\nolimits\!\left(x\right)\mathop{Q_{{k}}\/}\nolimits\!\left(y\right)=(n+1)\left(\mathop{P_{{n+1}}\/}\nolimits\!\left(x\right)\mathop{Q_{{n}}\/}\nolimits\!\left(y\right)-\mathop{P_{{n}}\/}\nolimits\!\left(x\right)\mathop{Q_{{n+1}}\/}\nolimits\!\left(y\right)\right)-1.$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind and $n$ : nonnegative integer
A&S Ref:: 8.9.2
Permalink:: http://dlmf.nist.gov/14.18.E7
Encodings:: TeX, pMML, png

¶ Zonal Harmonic Series

14.18.8 $\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(-x\right)=\frac{\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)}{\pi}\sum _{{n=0}}^{{\infty}}\frac{2n+1}{(\nu-n)(\nu+n+1)}\mathop{\mathsf{P}_{{n}}\/}\nolimits\!\left(x\right),$ $\nu\notin\Integer$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\Integer$ : set of all integers, $\notin$ : not an element of, $\mathop{\sin\/}\nolimits z$ : sine function, $n$ : nonnegative integer and $\nu$ : general degree
Referenced by:: §14.18(iii)
Permalink:: http://dlmf.nist.gov/14.18.E8
Encodings:: TeX, pMML, png

¶ Dougall’s Expansion

Keywords:: Dougall’s expansion, associated Legendre functions

14.18.9 $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)}{\pi}\sum _{{n=0}}^{{\infty}}(-1)^{n}\frac{2n+1}{(\nu-n)(\nu+n+1)}\mathop{\mathsf{P}^{{-\mu}}_{{n}}\/}\nolimits\!\left(x\right),$ $-1<x\leq 1$ , $\mu\geq 0$ , $\nu\notin\Integer$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\Integer$ : set of all integers, $\notin$ : not an element of, $\mathop{\sin\/}\nolimits z$ : sine function, $n$ : nonnegative integer, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.18(iii)
Permalink:: http://dlmf.nist.gov/14.18.E9
Encodings:: TeX, pMML, png

For a series representation of the Dirac delta in terms of products of Legendre polynomials see (1.17.22).

§14.18(iv) Compendia

Keywords:: Ferrers functions, associated Legendre functions
Referenced by:: §14.28(iii)
Permalink:: http://dlmf.nist.gov/14.18.iv

For collections of sums involving associated Legendre functions, see Hansen (1975, pp. 367–377, 457–460, and 475), Erdélyi et al. (1953a, §3.10), Gradshteyn and Ryzhik (2000, §8.92), Magnus et al. (1966, pp. 178–184), and Prudnikov et al. (1990, §§5.2, 6.5). See also §18.18 and (34.3.19).