§13.13 Addition and Multiplication Theorems

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

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

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