13 Confluent Hypergeometric FunctionsKummer Functions13.12 Products13.14 Definitions and Basic Properties

§13.13 Addition and Multiplication Theorems

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Referenced by:

§13.11

Permalink:

http://dlmf.nist.gov/13.13

See also:

Annotations for Ch.13

Contents

1. §13.13(i) Addition Theorems for $M(a,b,z)$
2. §13.13(ii) Addition Theorems for $U(a,b,z)$
3. §13.13(iii) Multiplication Theorems for $M(a,b,z)$ and $U(a,b,z)$

§13.13(i) Addition Theorems for $M(a,b,z)$

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Keywords:

Kummer functions, addition theorems, series expansions

Notes:

See Erdélyi et al. (1953a, §6.14) and Slater (1960, §2.3). In the first reference Equation (2) needs the constraint and Equation (6) should have no constraint. In the second reference (2.3.6) contains an error: ${(x+y)}^n$ should be replaced by ${(x+y)}^{-n}$.

Permalink:

http://dlmf.nist.gov/13.13.i

See also:

Annotations for §13.13 and Ch.13

The function $M(a,b,x+y)$ has the following expansions:

13.13.1		$$\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_n{y}^n}{{\left(b\right)}_{n}n!}M(a+n,b+n,x),$$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/13.13.E1 Encodings: TeX, pMML, png See also: Annotations for §13.13(i), §13.13 and Ch.13

13.13.2		$${\left(\frac{x+y}{x}\right)}^{1-b}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-b\right)}_n{(-y/x)}^n}{n!}M(a,b-n,x),$$
,
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/13.13.E2 Encodings: TeX, pMML, png See also: Annotations for §13.13(i), §13.13 and Ch.13

13.13.3		$${\left(\frac{x}{x+y}\right)}^a\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_n{y}^n}{n!{(x+y)}^n}M(a+n,b,x),$$
$\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$,
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/13.13.E3 Encodings: TeX, pMML, png See also: Annotations for §13.13(i), §13.13 and Ch.13

13.13.4		$${\mathrm{e}}^y\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(b-a\right)}_n{(-y)}^n}{{\left(b\right)}_{n}n!}M(a,b+n,x),$$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/13.13.E4 Encodings: TeX, pMML, png See also: Annotations for §13.13(i), §13.13 and Ch.13

13.13.5		$${\mathrm{e}}^y{\left(\frac{x}{x+y}\right)}^{b-a}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(b-a\right)}_n{y}^n}{n!{(x+y)}^n}M(a-n,b,x),$$
$\mathrm{\Re }\left((y+x)/x\right)>\frac{1}{2}$,
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/13.13.E5 Encodings: TeX, pMML, png See also: Annotations for §13.13(i), §13.13 and Ch.13

13.13.6		$${\mathrm{e}}^y{\left(\frac{x+y}{x}\right)}^{1-b}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-b\right)}_n{(-y)}^n}{n!x^n}M(a-n,b-n,x),$$
.
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/13.13.E6 Encodings: TeX, pMML, png See also: Annotations for §13.13(i), §13.13 and Ch.13

§13.13(ii) Addition Theorems for $U(a,b,z)$

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Keywords:

Kummer functions, addition theorems, series expansions

Notes:

See Slater (1960, §2.3.1).

Permalink:

http://dlmf.nist.gov/13.13.ii

See also:

Annotations for §13.13 and Ch.13

The function $U(a,b,x+y)$ has the following expansions:

13.13.7		$$\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_n{(-y)}^n}{n!}U(a+n,b+n,x),$$
,
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/13.13.E7 Encodings: TeX, pMML, png See also: Annotations for §13.13(ii), §13.13 and Ch.13

13.13.8		$${\left(\frac{x+y}{x}\right)}^{1-b}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1+a-b\right)}_n{(-y/x)}^n}{n!}U(a,b-n,x),$$
,
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/13.13.E8 Encodings: TeX, pMML, png See also: Annotations for §13.13(ii), §13.13 and Ch.13

13.13.9		$${\left(\frac{x}{x+y}\right)}^a\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_n{\left(1+a-b\right)}_n{y}^n}{n!{(x+y)}^n}U(a+n,b,x),$$
$\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$,
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/13.13.E9 Encodings: TeX, pMML, png See also: Annotations for §13.13(ii), §13.13 and Ch.13

13.13.10		$${\mathrm{e}}^y\sum _{n=0}^{\mathrm{\infty }}\frac{{(-y)}^n}{n!}U(a,b+n,x),$$
,
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/13.13.E10 Encodings: TeX, pMML, png See also: Annotations for §13.13(ii), §13.13 and Ch.13

13.13.11		$${\mathrm{e}}^y{\left(\frac{x}{x+y}\right)}^{b-a}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-y)}^n}{n!{(x+y)}^n}U(a-n,b,x),$$
$\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$,
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/13.13.E11 Encodings: TeX, pMML, png See also: Annotations for §13.13(ii), §13.13 and Ch.13

13.13.12		$${\mathrm{e}}^y{\left(\frac{x+y}{x}\right)}^{1-b}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-y)}^n}{n!x^n}U(a-n,b-n,x),$$
.
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/13.13.E12 Encodings: TeX, pMML, png See also: Annotations for §13.13(ii), §13.13 and Ch.13

§13.13(iii) Multiplication Theorems for $M(a,b,z)$ and $U(a,b,z)$

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Keywords:

Kummer functions, multiplication theorems, series expansions

Notes:

See Slater (1960, §§2.3.2, 2.3.3).

Permalink:

http://dlmf.nist.gov/13.13.iii

See also:

Annotations for §13.13 and Ch.13

To obtain similar expansions for $M(a,b,xy)$ and $U(a,b,xy)$, replace $y$ in the previous two subsections by $(y-1)x$.

13.12 Products13.14 Definitions and Basic Properties

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