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An extension of the Hille-Hardy formula


Author: H. M. Srivastava
Journal: Math. Comp. 23 (1969), 305-311
MSC: Primary 33.20
DOI: https://doi.org/10.1090/S0025-5718-1969-0243132-4
MathSciNet review: 0243132
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Abstract: While attempting to give extensions of the well-known Hille-Hardy formula for the generalized Laguerre polynomials $ \{ {L_n}^{(\alpha )}(x)\} $ defined by

$\displaystyle {(1 - t)^{ - 1 - \alpha }}\exp \left[ { - \frac{{xt}} {{1 - t}}} \right] = \sum\limits_{n = 0}^\infty {{L_n}^{(\alpha )}} (x){t^n}$

, the author applies here certain operational techniques and the method of finite mathematical induction to derive several bilinear generating functions associated with various classes of generalized hypergeometric polynomials. It is observed that the earlier works of Brafman [2], [3], [4], Chaundy [5], Meixner [12], Weisner [16], and others quoted in the literature, are only specialized or limiting forms of the results presented here.

References [Enhancements On Off] (What's this?)

  • [1] N. Abdul-Halim & W. A. Al-Salam, ``Double Euler transformations of certain hypergeometric functions,'' Duke Math. J., v. 30, 1963, pp. 51-62. MR 26 #1500. MR 0143951 (26:1500)
  • [2] F. Brafman, ``Generating functions of Jacobi and related polynomials,'' Proc. Amer. Math. Soc., v. 2, 1951, pp. 942-949. MR 13, 649. MR 0045875 (13:649i)
  • [3] F. Brafman, ``Some generating functions for Laguerre and Hermite polynomials,'' Canad. J. Math., v. 9, 1957, pp. 180-187. MR 19, 28. MR 0085363 (19:28b)
  • [4] F. Brafman, ``An ultraspherical generating function,'' Pacific J. Math., v. 7, 1957, pp. 1319-1323. MR 21 #1410. MR 0102620 (21:1410)
  • [5] T. W. Chaundy, ``An extension of hypergeometric functions,'' Quart. J. Math. Oxford Ser., v. 14, 1943, pp. 55-78. MR 6, 64. MR 0010749 (6:64d)
  • [6] A. Erdélyi, ``Funktionalrelationen mit konfluenten hypergeometrischen Funktionen. II: Reihenentwicklungen,'' Math. Z., v. 42, 1937, pp. 641-670.
  • [7] A. Erdélyi, et al., Higher Transcendental Functions. Vol. I, McGraw-Hill, New York, 1953. MR 15, 419.
  • [8] A. Erdélyi, et al., Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953. MR 15, 419.
  • [9] A. Erdélyi, et al., Tables of Integral Transforms, Vol. I, McGraw-Hill, New York, 1954. MR 15, 868. MR 0061695 (15:868a)
  • [10] G. H. Hardy, ``Summation of a series of polynomials of Laguerre,'' J. London Math. Soc., v. 7, 1932, pp. 138-139; addendum, 192.
  • [11] E. Hille, ``On Laguerre's series. I, II, III,'' Proc. Nat. Acad. Sci. U.S.A., v. 12, 1926, pp. 261-269; 348-352.
  • [12] J. Meixner, ``Umformung gewisser Reihen, deren Glieder Produkte hypergeometrischer Funktionen sind,'' Deutsche Math., v. 6, 1942, pp. 341-349. MR 4, 275. MR 0008287 (4:275b)
  • [13] H. M. Srivastava & C. M. Joshi, ``Certain double Whittaker transforms of generalized hypergeometric functions,'' Yokohama Math. J., v. 15, 1967, pp. 17-32. MR 0234224 (38:2542)
  • [14] H. M. Srivastava & J. P. Singhal, ``Double Meijer transformations of certain hypergeometric functions,'' Proc. Cambridge Philos. Soc., v. 64, 1968, pp. 425-430. MR 36 #6660. MR 0223612 (36:6660)
  • [15] G. N. Watson, ``Notes on generating functions of polynomials. I: Laguerre polynomials,'' J. London Math. Soc., v. 8, 1933, pp. 189-192.
  • [16] L. Weisner, ``Group-theoretic origin of certain generating functions,'' Pacific J. Math., v. 5, 1955, pp. 1033-1039. MR 19, 264. MR 0086905 (19:264e)

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DOI: https://doi.org/10.1090/S0025-5718-1969-0243132-4
Article copyright: © Copyright 1969 American Mathematical Society

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