# Mehler functions

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##### 1: 14.1 Special Notation
The main functions treated in this chapter are the Legendre functions $\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}_{\nu}\left(x\right)$, $P_{\nu}\left(z\right)$, $Q_{\nu}\left(z\right)$; Ferrers functions $\mathsf{P}^{\mu}_{\nu}\left(x\right)$, $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P^{\mu}_{\nu}\left(z\right)$, $Q^{\mu}_{\nu}\left(z\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$; conical functions $\mathsf{P}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $\mathsf{Q}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ (also known as Mehler functions). …
##### 2: 14.31 Other Applications
###### §14.31(ii) Conical Functions
These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …
##### 4: 14.20 Conical (or Mehler) Functions
###### §14.20 Conical (or Mehler) Functions
Solutions are known as conical or Mehler functions. …
##### 7: 18.11 Relations to Other Functions
###### §18.11 Relations to Other Functions
See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions.
##### 8: 18.18 Sums
###### Hermite
Formula (18.18.28) is known as the Mehler formula. …
##### 9: 18.10 Integral Representations
###### §18.10(i) Dirichlet–Mehler-Type Integral Representations
18.10.5 $P_{n}\left(\cos\theta\right)=\frac{1}{\pi}\int_{0}^{\pi}(\cos\theta+i\sin% \theta\cos\phi)^{n}\,\mathrm{d}\phi.$
18.10.6 $L^{(\alpha)}_{n}\left(x^{2}\right)=\frac{2(-1)^{n}}{{\pi}^{\frac{1}{2}}\Gamma% \left(\alpha+\tfrac{1}{2}\right)n!}\*\int_{0}^{\infty}\int_{0}^{\pi}{(x^{2}-r^% {2}+2ixr\cos\phi)^{n}}\*{\mathrm{e}}^{-r^{2}}r^{2\alpha+1}(\sin\phi)^{2\alpha}% \,\mathrm{d}\phi\,\mathrm{d}r,$ $\alpha>-\frac{1}{2}$.
###### Laguerre
For the Bessel function $J_{\nu}\left(z\right)$ see §10.2(ii). …
##### 10: Bibliography G
• G. Gasper (1975) Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.
• A. Gervois and H. Navelet (1985a) Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.
• A. Gervois and H. Navelet (1985b) Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.
• S. Goldstein (1927) Mathieu functions. Trans. Camb. Philos. Soc. 23, pp. 303–336.
• A. J. Guttmann and T. Prellberg (1993) Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.