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Mehler–Heine type formulas

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1: 18.11 Relations to Other Functions
§18.11(i) Explicit Formulas
§18.11(ii) Formulas of MehlerHeine Type
2: 14.12 Integral Representations
§14.12(i) - 1 < x < 1
Mehler–Dirichlet Formula
Heine’s Integral
3: 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: 14.28 Sums
§14.28(ii) Heine’s Formula
5: 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.
  • A. B. Olde Daalhuis and N. M. Temme (1994) Uniform Airy-type expansions of integrals. SIAM J. Math. Anal. 25 (2), pp. 304–321.
  • 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.
  • 6: 18.10 Integral Representations
    §18.10(i) Dirichlet–Mehler-Type Integral Representations
    §18.10(ii) Laplace-Type Integral Representations
    7: Bibliography G
  • G. Gasper (1972) An inequality of Turán type for Jacobi polynomials. Proc. Amer. Math. Soc. 32, pp. 435–439.
  • 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.
  • W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
  • A. Gil, J. Segura, and N. M. Temme (2003a) Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion. J. Comput. Appl. Math. 153 (1-2), pp. 225–234.
  • D. P. Gupta and M. E. Muldoon (2000) Riccati equations and convolution formulae for functions of Rayleigh type. J. Phys. A 33 (7), pp. 1363–1368.
  • 8: 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
    9: Bibliography V
  • A. J. van der Poorten (1980) Some Wonderful Formulas an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.
  • A. Verma and V. K. Jain (1983) Certain summation formulae for q -series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).
  • R. Vidūnas and N. M. Temme (2002) Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.
  • H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.
  • 10: 14.1 Special Notation
    The main functions treated in this chapter are the Legendre functions P ν ( x ) , Q ν ( x ) , P ν ( z ) , Q ν ( z ) ; Ferrers functions P ν μ ( x ) , Q ν μ ( x ) (also known as the Legendre functions on the cut); associated Legendre functions P ν μ ( z ) , Q ν μ ( z ) , Q ν μ ( z ) ; conical functions P - 1 2 + i τ μ ( x ) , Q - 1 2 + i τ μ ( x ) , Q ^ - 1 2 + i τ μ ( x ) , P - 1 2 + i τ μ ( x ) , Q - 1 2 + i τ μ ( x ) (also known as Mehler functions). …