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Mehler formula

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1: 14.12 Integral Representations
§14.12(i) 1 < x < 1
Mehler–Dirichlet Formula
2: 18.11 Relations to Other Functions
§18.11(ii) Formulas of Mehler–Heine Type
3: 18.18 Sums
Hermite
Formula (18.18.28) is known as the Mehler formula. …
4: 18.7 Interrelations and Limit Relations
See §18.11(ii) for limit formulas of Mehler–Heine type.
5: 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.
  • 6: 14.20 Conical (or Mehler) Functions
    §14.20 Conical (or Mehler) Functions
    Solutions are known as conical or Mehler functions. …
    14.20.2 𝖰 ^ 1 2 + i τ μ ( x ) = ( e μ π i 𝖰 1 2 + i τ μ ( x ) ) 1 2 π sin ( μ π ) 𝖯 1 2 + i τ μ ( x ) .
    14.20.6 P 1 2 + i τ μ ( x ) = i e μ π i sinh ( τ π ) | Γ ( μ + 1 2 + i τ ) | 2 ( Q 1 2 + i τ μ ( x ) Q 1 2 i τ μ ( x ) ) , τ 0 .
    §14.20(vi) Generalized Mehler–Fock Transformation
    7: Bibliography O
  • F. Oberhettinger and T. P. Higgins (1961) Tables of Lebedev, Mehler and Generalized Mehler Transforms. Mathematical Note Technical Report 246, Boeing Scientific Research Lab, Seattle.
  • F. Oberhettinger (1973) Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.
  • F. W. J. Olver (1977a) Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.
  • F. W. J. Olver (1977b) Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.
  • F. W. J. Olver (1978) General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.