§15.2 Definitions and Analytical Properties

The hypergeometric function is defined by the Gauss series

15.2.1

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Defines:

or : Gauss’ hypergeometric function

Symbols:

: gamma function, : notation for Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter

A&S Ref:

15.1.1

Referenced by:

§15.19(i), §15.4(iii), §18.30(v), §18.35(i), §18.35(ii), §19.19

Permalink:

http://dlmf.nist.gov/15.2.E1

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Annotations for §15.2(i), §15.2 and Ch.15

on the disk , and by analytic continuation elsewhere. In general, does not exist when . The branch obtained by introducing a cut from 1 to on the real -axis, that is, the branch in the sector , is the principal branch (or principal value) of .

For all values of

15.2.2 ,

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Defines:

or : Olver’s hypergeometric function

Symbols:

: gamma function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), or : scaled (or Olver’s) generalized hypergeometric function, : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter

Referenced by:

§15.12(i), §16.2(v)

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http://dlmf.nist.gov/15.2.E2

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Annotations for §15.2(i), §15.2 and Ch.15

again with analytic continuation for other values of , and with the principal branch defined in a similar way.

Except where indicated otherwise principal branches of and are assumed throughout the DLMF.

The difference between the principal branches on the two sides of the branch cut (§4.2(i)) is given by

15.2.3 .

ⓘ

Symbols:

: gamma function, : the ratio of the circumference of a circle to its diameter, : imaginary unit, or : Olver’s hypergeometric function, : real variable, : real or complex parameter, : real or complex parameter and : real or complex parameter

Referenced by:

§15.2(i)

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On the circle of convergence, , the Gauss series:

• (a)

  Converges absolutely when .

• (b)

  Converges conditionally when and is excluded.

• (c)

  Diverges when .

For the case see also §15.4(ii).

The principal branch of is an entire function of , , and . The same is true of other branches, provided that , 1, and are excluded. As a multivalued function of , is analytic everywhere except for possible branch points at , 1, and . The same properties hold for , except that as a function of , in general has poles at .

Because of the analytic properties with respect to , , and , it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. In particular

15.2.3_5 .

ⓘ

Symbols:

: gamma function, or : Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), or : Olver’s hypergeometric function, : integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter

Referenced by:

§15.2(ii), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)

Permalink:

http://dlmf.nist.gov/15.2.E3_5

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Clarification (effective with 1.0.28):

This equation was given a number.

See also:

Annotations for §15.2(ii), §15.2 and Ch.15

For example, when , , and , is a polynomial:

15.2.4

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Symbols:

or : Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : binomial coefficient, : factorial (as in ), : integer, : complex variable, : real or complex parameter, : real or complex parameter and : integer

A&S Ref:

15.4.1

Permalink:

http://dlmf.nist.gov/15.2.E4

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Annotations for §15.2(ii), §15.2 and Ch.15

This formula is also valid when , , provided that we use the interpretation

15.2.5

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Symbols:

or : Gauss’ hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : integer and : integer

Referenced by:

§15.10(i), §15.2(ii), §15.2(ii), §15.4(i), §15.4(iii)

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http://dlmf.nist.gov/15.2.E5

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and not

15.2.6

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Symbols:

or : Gauss’ hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : integer and : integer

Referenced by:

§15.2(ii), §15.2(ii), §15.4(i), §15.4(iii), Erratum (V1.0.16) for Subsection 15.2(ii)

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http://dlmf.nist.gov/15.2.E6

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which sometimes needs to be used in §15.4. (Both interpretations give solutions of the hypergeometric differential equation (15.10.1), as does , which is analytic at .)

For comparison of and , with the former using the limit interpretation (15.2.5), see Figures 15.3.6 and 15.3.7.

Let be a nonnegative integer. Formula (15.4.6) reads . The right-hand side can be seen as an analytical continuation for the left-hand side when approaches . In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that is equal to the first terms of the Maclaurin series for .