View source for Confluent hypergeometric function

''Kummer function, Pochhammer function''

A solution of the [[Confluent hypergeometric equation|confluent hypergeometric equation]]

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The function may be defined using the so-called Kummer series

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247005.png" /> are parameters which assume any real or complex values except for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247007.png" /> is a complex variable. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247008.png" /> is called the confluent hypergeometric function of the first kind. The second linearly independent solution of equation (1),

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is called the confluent hypergeometric function of the second kind.

The confluent hypergeometric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470011.png" /> is an entire analytic function in the entire complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470012.png" />-plane; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470013.png" /> is fixed, it is an entire function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470014.png" /> and a meromorphic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470015.png" /> with simple poles at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470016.png" />. The confluent hypergeometric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470017.png" /> is an analytic function in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470018.png" />-plane with the slit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470019.png" /> and an entire function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470021.png" />.

The confluent hypergeometric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470022.png" /> is connected with the [[Hypergeometric function|hypergeometric function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470023.png" /> by the relation

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Elementary relationships. The four functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470026.png" /> are called adjacent (or contiguous) to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470027.png" />. There is a linear relationship between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470028.png" /> and any two functions adjacent to it, e.g.

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Six formulas of this type may be obtained from the relations between adjacent functions for hypergeometric functions. The successive use of these recurrence formulas yields linear relations connecting the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470030.png" /> with the associated functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470033.png" /> are integers.

Differentiation formulas:

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Basic integral representations.

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The asymptotic behaviour of confluent hypergeometric functions as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470040.png" /> can be studied using the integral representations [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470041.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470043.png" /> are bounded, the behaviour of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470044.png" /> is described by formula (2). In particular, for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470045.png" /> and bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470047.png" />:

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Representations of functions by confluent hypergeometric functions.

Bessel functions:

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Laguerre polynomials:

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Probability integrals:

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The exponential integral function:

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The [[logarithmic integral]] function:

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Gamma-functions:

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Elementary functions:

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See also [[#References|[1]]], [[#References|[2]]], [[#References|[3]]], [[#References|[8]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.S. Gradshtein,   I.M. Ryzhik,   "Table of integrals, series and products" , Acad. Press  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Abramowitz,   I.A. Stegun,   "Handbook of mathematical functions" , Dover, reprint  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.T. Whittaker,   G.N. Watson,   "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 6</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.L. Lebedev,   R.M. Fedorova,   "Handbook of mathematical tables" , Moscow  (1956)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N.M. Burunova,   "Handbook of mathematical tables" , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.A. Fletcher,   J.C.P. Miller,   L. Rosenhead,   L.J. Comrie,   "An index of mathematical tables" , '''1–2''' , Oxford Univ. Press  (1962)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  N.N. Lebedev,   "Special functions and their applications" , Prentice-Hall  (1965)  (Translated from Russian)</TD></TR></table>