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Series expansions of $ W\sb{k,\,m}(Z)$ involving parabolic cylinder functions

Authors: R. Wong and E. Rosenbloom
Journal: Math. Comp. 25 (1971), 783-787
MSC: Primary 33A30
MathSciNet review: 0306566
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Abstract: In this paper, an explicit error bound is obtained for an expansion of the Whittaker function, $ {W_{k,m}}(z)$, in series of parabolic cylinder functions. It is also shown that the Whittaker function may be asymptotically represented as the sum of two products where one product involves a parabolic cylinder function and the other product involves the first-order derivative of this function.

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  • [1] Herbert Buchholz, Die konfluente hypergeometrische Funktion mit besonderer Berücksichtigung ihrer Anwendungen, Ergebnisse der angewandten Mathematik. Bd. 2, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). MR 0054783
  • [2] A. Erdélyi, W. Magnus, F. Oberhettinger & F. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953. MR 15. 419.
  • [3] Artur Erdélyi, Über eine Integraldarstellung der 𝑊^{𝑘,𝑚}-Funktionen und ihre Darstellung durch die Funktionen des parabolischen Zylinders, Math. Ann. 113 (1937), no. 1, 347–356 (German). MR 1513095, 10.1007/BF01571638
  • [4] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
  • [5] R. Wong, On uniform asymptotic expansion of definite integrals, J. Approximation Theory 7 (1973), 76–86. MR 0340910

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Keywords: Whittaker function, parabolic cylinder functions, error bound, asymptotic representation
Article copyright: © Copyright 1971 American Mathematical Society