Mehler functions

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1: 14.1 Special Notation

… ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). …

2: 14.31 Other Applications

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§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. …

3: 14.12 Integral Representations

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§14.12(i)

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4: 14.20 Conical (or Mehler) Functions

§14.20 Conical (or Mehler) Functions

… ►Solutions are known as conical or Mehler functions. … ►

§14.20(vi) Generalized Mehler–Fock Transformation

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5: 14.34 Software

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§14.34(iv) Conical (Mehler) and/or Toroidal Functions

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6: 10.9 Integral Representations

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Mehler–Sonine and Related Integrals

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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. ►

Ultraspherical

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Hermite

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§18.11(ii) Formulas of Mehler–Heine Type

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8: 37.17 Hermite Polynomials on ${ℝ}^d$

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Generating Functions

… ►Specialization in §37.13(i) of the rotation invariant weight function to $W(𝐱)=\exp \left(-{\Vert 𝐱\Vert }^2\right)$ gives for the corresponding OPs that … ►

§37.17(iv) Mehler Formula

►The Poisson kernel (37.13.6) of ${𝒱}_n({ℝ}^d)$ is given explicitly by the Mehler formula … ►The basis functions (37.17.2) and (37.17.8) of $𝒱({ℝ}^d)$ are limits of the basis functions (37.15.5) and (37.15.7) of ${𝒱}_n^{\alpha }({𝔹}^d)$: …

9: 18.18 Sums

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§18.18(i) Series Expansions of Arbitrary Functions

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Expansion of $L^2$ functions

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Laguerre

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Hermite

… ►Formula (18.18.28) is known as the Mehler formula. …

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