Expansion of the confluent hypergeometric function in series of Bessel functions

Author:
Yudell L. Luke

Journal:
Math. Comp. **13** (1959), 261-271

MSC:
Primary 33.00; Secondary 65.00

DOI:
https://doi.org/10.1090/S0025-5718-1959-0107027-2

MathSciNet review:
0107027

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Abstract: An expansion of the confluent hypergeometric function in series of functions of the same kind has been given by Buchholz [1]. By specialization of some quantities, there is obtained an expansion in series of modified Bessel functions of the first kind, , where depends on the parameter . Tricomi [2, 3] has developed two expansions of similar type where both the order and argument of the Bessel functions depend on the parameters and . In the present paper, we derive an expansion in series of Bessel functions of integral order whose argument is independent of and . Our expansion is advantageous for many purposes of computation since the parameters and variable of appear in separated form. Also, for desk calculation, extensive tables of are available, while for automatic computation Bessel functions are easy to generate [4].

Special cases of the confluent function, such as the incomplete gamma function, are also studied. For the class of transcendents known as the error functions, including the Fresnel integrals, it is shown that our expansion coincides with that of Buchholz [1]. By specializing a parameter and passing to a limit, we derive expansions for the exponential integral and related functions. Other expansions for the error and exponential integrals are derived on altogether different bases. Finally, some numerical examples are presented to manifest the efficiency of our formulas.

**[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]**Francesco G. Tricomi,*Sul comportamento asintotico dei polinomi di Laguerre*, Ann. Mat. Pura Appl. (4)**28**(1949), 263–289 (Italian). MR**0036355**, https://doi.org/10.1007/BF02411134**[3]**Francesco G. Tricomi,*Sulla funzione gamma incompleta*, Ann. Mat. Pura Appl. (4)**31**(1950), 263–279 (Italian). MR**0047834**, https://doi.org/10.1007/BF02428264**[4]**Irene A. Stegun and Milton Abramowitz,*Generation of Bessel functions on high speed computers*, Math. Tables Aids Comput**11**(1957), 255–257. MR**0093939**, https://doi.org/10.1090/S0025-5718-1957-0093939-3**[5]**Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,*Higher transcendental functions. Vols. I, II*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR**0058756****[6]**G. N. Watson,*A treatise on the theory of Bessel functions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR**1349110****[7]**H.T.F., v. 1, Ch. 2 and Ch. 4.**[8]**H.T.F., v. 1, p. 80.**[10]**H.T.F., v. 1, Ch. 1.**[11]**H.T.F., v. 1, p. 276-277. See also 2 and 3.**[12]**H. T. F., v. 2, p. 148.**[13]**A. N. Lowan, G. Blanch, & M. Abramowitz, ``Table of and related functions", in*Tables of Functions and Zeros of Functions*, National Bureau of Standards, AMS 37, November, 1954, p. 33-39.**[14]**P. R. Ansell, & R. A. Fisher, ``Note on the numerical evaluation of a Bessel function derivative", London Math. Soc.*Proc.*, v. 24, 1926, p. liv-lvi.**[15]**J. Airey, ``The Bessel function derivative and ,*Phil. Mag.*, v. 19, 1935, p. 236-243.**[16]**F. Oberhettinger,*On the derivative of Bessel functions with respect to the order*, J. Math. and Phys.**37**(1958), 75–78. MR**0092871**, https://doi.org/10.1002/sapm195837175**[17]**See 6, p. 143.**[18]**A. N. Lowan & W. Horenstein, ``On the function ", in*Tables of Functions and of Zeros of Functions*, National Bureau of Standards, AMS 37, November, 1954, p. 1-20.

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DOI:
https://doi.org/10.1090/S0025-5718-1959-0107027-2

Article copyright:
© Copyright 1959
American Mathematical Society