View source for Incomplete beta-function

The function defined by the formula

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where

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is the [[Beta-function|beta-function]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050460/i0504604.png" /> is an integer, then

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Series representation:

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Continued fraction representation:

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where

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Asymptotic representation for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050460/i05046012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050460/i05046013.png" />:

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where

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Asymptotic representation for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050460/i05046017.png" /> and bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050460/i05046018.png" />:

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where

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Connection with the [[Hypergeometric function|hypergeometric function]]:

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Recurrence relations:

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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Abramowitz,   I.A. Stegun,   "Handbook of mathematical functions" , Dover, reprint  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Pearson,   "Tables of the incomplete beta-function" , Cambridge Univ. Press  (1932)</TD></TR></table>