# §13.2 Definitions and Basic Properties

## §13.2(i) Differential Equation

### Kummer’s Equation

 13.2.1 $z\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(b-z)\frac{\mathrm{d}w}{\mathrm{d% }z}-aw=0.$ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ and $z$: complex variable A&S Ref: 13.1.1 Referenced by: §13.14(i), §13.14(v), §13.2(i), §13.2(v), §13.29(ii), §13.3(i), §13.3(i) Permalink: http://dlmf.nist.gov/13.2.E1 Encodings: TeX, pMML, png See also: Annotations for 13.2(i)

This equation has a regular singularity at the origin with indices $0$ and $1-b$, and an irregular singularity at infinity of rank one. It can be regarded as the limiting form of the hypergeometric differential equation (§15.10(i)) that is obtained on replacing $z$ by $\ifrac{z}{b}$, letting $b\to\infty$, and subsequently replacing the symbol $c$ by $b$. In effect, the regular singularities of the hypergeometric differential equation at $b$ and $\infty$ coalesce into an irregular singularity at $\infty$.

### Standard Solutions

The first two standard solutions are:

 13.2.2 $\mathop{M\/}\nolimits\!\left(a,b,z\right)=\sum_{s=0}^{\infty}\frac{{\left(a% \right)_{s}}}{{\left(b\right)_{s}}s!}z^{s}=1+\frac{a}{b}z+\frac{a(a+1)}{b(b+1)% 2!}z^{2}+\cdots,$ Defines: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function Symbols: $\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$: $=\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer and $z$: complex variable A&S Ref: 13.1.2 Referenced by: §13.14(i), §13.2(i), §13.29(i), §13.29(ii), §13.9(i) Permalink: http://dlmf.nist.gov/13.2.E2 Encodings: TeX, pMML, png See also: Annotations for 13.2(i)

and

 13.2.3 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)=\sum_{s=0}^{\infty}\frac{% {\left(a\right)_{s}}}{\mathop{\Gamma\/}\nolimits\!\left(b+s\right)s!}z^{s},$ Defines: $\mathop{{\mathbf{M}}\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Olver’s confluent hypergeometric function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer and $z$: complex variable Referenced by: §13.2(i) Permalink: http://dlmf.nist.gov/13.2.E3 Encodings: TeX, pMML, png See also: Annotations for 13.2(i)

except that $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ does not exist when $b$ is a nonpositive integer. In other cases

 13.2.4 $\mathop{M\/}\nolimits\!\left(a,b,z\right)=\mathop{\Gamma\/}\nolimits\!\left(b% \right)\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right).$

The series (13.2.2) and (13.2.3) converge for all $z\in\mathbb{C}$. $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ is entire in $z$ and $a$, and is a meromorphic function of $b$. $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ is entire in $z$, $a$, and $b$.

Although $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ does not exist when $b=-n$, $n=0,1,2,\dots$, many formulas containing $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ continue to apply in their limiting form. In particular,

 13.2.5 $\lim_{b\to-n}\frac{\mathop{M\/}\nolimits\!\left(a,b,z\right)}{\mathop{\Gamma\/% }\nolimits\!\left(b\right)}=\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,-n,z% \right)=\frac{{\left(a\right)_{n+1}}}{(n+1)!}z^{n+1}\mathop{M\/}\nolimits\!% \left(a+n+1,n+2,z\right).$

When $a=-n$, $n=0,1,2,\dots$, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ is a polynomial in $z$ of degree not exceeding $n$; this is also true of $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ provided that $b$ is not a nonpositive integer.

Another standard solution of (13.2.1) is $\mathop{U\/}\nolimits\!\left(a,b,z\right)$, which is determined uniquely by the property

 13.2.6 $\mathop{U\/}\nolimits\!\left(a,b,z\right)\sim z^{-a},$ $z\to\infty$, $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{3}{2}\pi-\delta$, Defines: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function Symbols: $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\mathrm{ph}\/}\nolimits$: phase, $z$: complex variable and $\delta$: small positive constant Referenced by: §13.2(i), §13.2(iv) Permalink: http://dlmf.nist.gov/13.2.E6 Encodings: TeX, pMML, png See also: Annotations for 13.2(i)

where $\delta$ is an arbitrary small positive constant. In general, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ has a branch point at $z=0$. The principal branch corresponds to the principal value of $z^{-a}$ in (13.2.6), and has a cut in the $z$-plane along the interval $(-\infty,0]$; compare §4.2(i).

When $a=-m$, $m=0,1,2,\dots$, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ is a polynomial in $z$ of degree $m$:

 13.2.7 $\mathop{U\/}\nolimits\!\left(-m,b,z\right)=(-1)^{m}{\left(b\right)_{m}}\mathop% {M\/}\nolimits\!\left(-m,b,z\right)=(-1)^{m}\sum_{s=0}^{m}\binom{m}{s}{\left(b% +s\right)_{m-s}}(-z)^{s}.$ Symbols: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\binom{\NVar{m}}{\NVar{n}}$: binomial coefficient, $m$: integer, $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable Referenced by: §13.2(i), Equation (13.2.7) Permalink: http://dlmf.nist.gov/13.2.E7 Encodings: TeX, pMML, png Addition (effective with 1.0.10): The equality $\mathop{U\/}\nolimits\!\left(-m,b,z\right)=(-1)^{m}{\left(b\right)_{m}}\mathop% {M\/}\nolimits\!\left(-m,b,z\right)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation $a=-n$ has been changed to $a=-m$. Reported 2015-02-10 See also: Annotations for 13.2(i)

Similarly, when $a-b+1=-n$, $n=0,1,2,\ldots$,

 13.2.8 $\mathop{U\/}\nolimits\!\left(a,a+n+1,z\right)=\frac{(-1)^{n}{\left(1-a-n\right% )_{n}}}{z^{a+n}}\mathop{M\/}\nolimits\!\left(-n,1-a-n,z\right)=z^{-a}\sum_{s=0% }^{n}\binom{n}{s}{\left(a\right)_{s}}z^{-s}.$ Symbols: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\binom{\NVar{m}}{\NVar{n}}$: binomial coefficient, $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable Referenced by: §13.2(i), Equation (13.2.8) Permalink: http://dlmf.nist.gov/13.2.E8 Encodings: TeX, pMML, png Addition (effective with 1.0.10): The equality $\mathop{U\/}\nolimits\!\left(a,a+n+1,z\right)=\frac{(-1)^{n}{\left(1-a-n\right% )_{n}}}{z^{a+n}}\mathop{M\/}\nolimits\!\left(-n,1-a-n,z\right)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Reported 2014-02-10 by Adri Olde Daalhuis See also: Annotations for 13.2(i)

When $b=n+1$, $n=0,1,2,\dots$, and $a\neq 0,-1,-2,\dots$,

 13.2.9 $\mathop{U\/}\nolimits\!\left(a,n+1,z\right)=\frac{(-1)^{n+1}}{n!\mathop{\Gamma% \/}\nolimits\!\left(a-n\right)}\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{% {\left(n+1\right)_{k}}k!}z^{k}\left(\mathop{\ln\/}\nolimits z+\mathop{\psi\/}% \nolimits\!\left(a+k\right)-\mathop{\psi\/}\nolimits\!\left(1+k\right)-\mathop% {\psi\/}\nolimits\!\left(n+k+1\right)\right)+\frac{1}{\mathop{\Gamma\/}% \nolimits\!\left(a\right)}\sum_{k=1}^{n}\frac{(k-1)!{\left(1-a+k\right)_{n-k}}% }{(n-k)!}z^{-k}.$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\mathop{\psi\/}\nolimits\!\left(\NVar{z}\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $n$: nonnegative integer and $z$: complex variable Referenced by: §13.2(i), §33.6, §6.6, Other Changes Permalink: http://dlmf.nist.gov/13.2.E9 Encodings: TeX, pMML, png Clarification (effective with 1.0.12): The condition $a\neq 0,-1,-2,\dots$ that appeared below this equation now appears ahead of the equation. Reported 2016-07-05 by Adri Olde Daalhuis See also: Annotations for 13.2(i)

When $b=n+1$, $n=0,1,2,\dots$, and $a=-m$, $m=0,1,2,\dots$,

 13.2.10 $\mathop{U\/}\nolimits\!\left(-m,n+1,z\right)=(-1)^{m}{\left(n+1\right)_{m}}% \mathop{M\/}\nolimits\!\left(-m,n+1,z\right)=(-1)^{m}\sum_{s=0}^{m}\binom{m}{s% }{\left(n+s+1\right)_{m-s}}(-z)^{s}.$ Symbols: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\binom{\NVar{m}}{\NVar{n}}$: binomial coefficient, $m$: integer, $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable Referenced by: §13.2(i), Other Changes, Equation (13.2.10) Permalink: http://dlmf.nist.gov/13.2.E10 Encodings: TeX, pMML, png Clarification (effective with 1.0.12): . The condition $a=-m,m=0,1,2,\dots$ that appeared below this equation now appears ahead of the equation. Reported 2016-07-05 by Adri Olde Daalhuis Addition (effective with 1.0.10): The equality $\mathop{U\/}\nolimits\!\left(-m,n+1,z\right)=(-1)^{m}{\left(n+1\right)_{m}}% \mathop{M\/}\nolimits\!\left(-m,n+1,z\right)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation $a=-m,m=0,1,2,\ldots$ has been introduced. Reported 2015-02-10 by Adri Olde Daalhuis See also: Annotations for 13.2(i)

When $b=-n$, $n=0,1,2,\dots$, the following equation can be combined with (13.2.9) and (13.2.10):

 13.2.11 $\mathop{U\/}\nolimits\!\left(a,-n,z\right)=z^{n+1}\mathop{U\/}\nolimits\!\left% (a+n+1,n+2,z\right).$

## §13.2(ii) Analytic Continuation

When $m\in\mathbb{Z}$,

 13.2.12 $\mathop{U\/}\nolimits\!\left(a,b,ze^{2\pi\mathrm{i}m}\right)=\frac{2\pi\mathrm% {i}e^{-\pi\mathrm{i}bm}\mathop{\sin\/}\nolimits\!\left(\pi bm\right)}{\mathop{% \Gamma\/}\nolimits\!\left(1+a-b\right)\mathop{\sin\/}\nolimits\!\left(\pi b% \right)}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)+e^{-2\pi\mathrm{i% }bm}\mathop{U\/}\nolimits\!\left(a,b,z\right).$

Except when $z=0$ each branch of $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ is entire in $a$ and $b$. Unless specified otherwise, however, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ is assumed to have its principal value.

## §13.2(iii) Limiting Forms as $z\to 0$

 13.2.13 $\mathop{M\/}\nolimits\!\left(a,b,z\right)=1+\mathop{O\/}\nolimits\!\left(z% \right).$

Next, in cases when $a=-n$ or $-n+b-1$, where $n$ is a nonnegative integer,

 13.2.14 $\mathop{U\/}\nolimits\!\left(-n,b,z\right)=(-1)^{n}{\left(b\right)_{n}}+% \mathop{O\/}\nolimits\!\left(z\right),$
 13.2.15 $\mathop{U\/}\nolimits\!\left(-n+b-1,b,z\right)=(-1)^{n}{\left(2-b\right)_{n}}z% ^{1-b}+\mathop{O\/}\nolimits\!\left(z^{2-b}\right).$

In all other cases

 13.2.16 $\displaystyle\mathop{U\/}\nolimits\!\left(a,b,z\right)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(b-1\right)}{\mathop{% \Gamma\/}\nolimits\!\left(a\right)}z^{1-b}+\mathop{O\/}\nolimits\!\left(z^{2-% \Re{b}}\right),$ $\Re{b}\geq 2$, $b\not=2$, Symbols: $\mathop{O\/}\nolimits\!\left(\NVar{x}\right)$: order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, $\Re{}$: real part and $z$: complex variable A&S Ref: 13.5.6 (with order estimate corrected) Permalink: http://dlmf.nist.gov/13.2.E16 Encodings: TeX, pMML, png See also: Annotations for 13.2(iii) 13.2.17 $\displaystyle\mathop{U\/}\nolimits\!\left(a,2,z\right)$ $\displaystyle=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}z^{-1}+% \mathop{O\/}\nolimits\!\left(\mathop{\ln\/}\nolimits z\right),$ 13.2.18 $\displaystyle\mathop{U\/}\nolimits\!\left(a,b,z\right)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(b-1\right)}{\mathop{% \Gamma\/}\nolimits\!\left(a\right)}z^{1-b}+\frac{\mathop{\Gamma\/}\nolimits\!% \left(1-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)}+\mathop{O\/}% \nolimits\!\left(z^{2-\Re{b}}\right),$ $1\leq\Re{b}<2$, $b\not=1$, 13.2.19 $\displaystyle\mathop{U\/}\nolimits\!\left(a,1,z\right)$ $\displaystyle=-\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}\left(% \mathop{\ln\/}\nolimits z+\mathop{\psi\/}\nolimits\!\left(a\right)+2\gamma% \right)+\mathop{O\/}\nolimits\!\left(z\mathop{\ln\/}\nolimits z\right),$ 13.2.20 $\displaystyle\mathop{U\/}\nolimits\!\left(a,b,z\right)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}{\mathop{% \Gamma\/}\nolimits\!\left(a-b+1\right)}+\mathop{O\/}\nolimits\!\left(z^{1-\Re{% b}}\right),$ $0<\Re{b}<1$, 13.2.21 $\displaystyle\mathop{U\/}\nolimits\!\left(a,0,z\right)$ $\displaystyle=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(a+1\right)}+\mathop{O% \/}\nolimits\!\left(z\mathop{\ln\/}\nolimits z\right),$ 13.2.22 $\displaystyle\mathop{U\/}\nolimits\!\left(a,b,z\right)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}{\mathop{% \Gamma\/}\nolimits\!\left(a-b+1\right)}+\mathop{O\/}\nolimits\!\left(z\right),$ $\Re{b}\leq 0$, $b\not=0$.

## §13.2(iv) Limiting Forms as $z\to\infty$

Except when $a=0,-1,\dots$ (polynomial cases),

 13.2.23 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)\sim\ifrac{e^{z}z^{a-b}}{% \mathop{\Gamma\/}\nolimits\!\left(a\right)},$ $\left|\mathop{\mathrm{ph}\/}\nolimits z\right|\leq\frac{1}{2}\pi-\delta$,

where $\delta$ is an arbitrary small positive constant.

For $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ see (13.2.6).

## §13.2(v) Numerically Satisfactory Solutions

Fundamental pairs of solutions of (13.2.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are

 13.2.24 $\mathop{U\/}\nolimits\!\left(a,b,z\right)$, $e^{z}\mathop{U\/}\nolimits\!\left(b-a,b,e^{-\pi\mathrm{i}}z\right)$, $-\tfrac{1}{2}\pi\leq\mathop{\mathrm{ph}\/}\nolimits{z}\leq\tfrac{3}{2}\pi$, 13.2.25 $\mathop{U\/}\nolimits\!\left(a,b,z\right)$, $e^{z}\mathop{U\/}\nolimits\!\left(b-a,b,e^{\pi\mathrm{i}}z\right)$, $-\tfrac{3}{2}\pi\leq\mathop{\mathrm{ph}\/}\nolimits{z}\leq\tfrac{1}{2}\pi$.

A fundamental pair of solutions that is numerically satisfactory near the origin is

 13.2.26 $\mathop{M\/}\nolimits\!\left(a,b,z\right),\quad z^{1-b}\mathop{M\/}\nolimits\!% \left(a-b+1,2-b,z\right),$ $b\not\in\mathbb{Z}$.

When $b=n+1=1,2,3,\dots$, a fundamental pair that is numerically satisfactory near the origin is $\mathop{M\/}\nolimits\!\left(a,n+1,z\right)$ and

 13.2.27 $\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(1-a\right)_{k}}}z^{-k}-\sum_{k=0}^{% \infty}\frac{{\left(a\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\mathop% {\ln\/}\nolimits z+\mathop{\psi\/}\nolimits\!\left(a+k\right)-\mathop{\psi\/}% \nolimits\!\left(1+k\right)-\mathop{\psi\/}\nolimits\!\left(n+k+1\right)\right),$

if $a-n\neq 0,-1,-2,\dots$, or $\mathop{M\/}\nolimits\!\left(a,n+1,z\right)$ and

 13.2.28 $\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(1-a\right)_{k}}}z^{-k}-\sum_{k=0}^{% -a}\frac{{\left(a\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\mathop{\ln% \/}\nolimits z+\mathop{\psi\/}\nolimits\!\left(1-a-k\right)-\mathop{\psi\/}% \nolimits\!\left(1+k\right)-\mathop{\psi\/}\nolimits\!\left(n+k+1\right)\right% )+(-1)^{1-a}(-a)!\sum_{k=1-a}^{\infty}\frac{(k-1+a)!}{{\left(n+1\right)_{k}}k!% }z^{k},$

if $a=0,-1,-2,\dots$, or $\mathop{M\/}\nolimits\!\left(a,n+1,z\right)$ and

 13.2.29 $\sum_{k=a}^{n}\frac{(k-1)!}{(n-k)!(k-a)!}z^{-k},$ Symbols: $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable Permalink: http://dlmf.nist.gov/13.2.E29 Encodings: TeX, pMML, png See also: Annotations for 13.2(v)

if $a=1,2,\dots,n$.

When $b=-n=0,-1,-2,\dots$, a fundamental pair that is numerically satisfactory near the origin is $z^{n+1}\*\mathop{M\/}\nolimits\!\left(a+n+1,n+2,z\right)$ and

 13.2.30 $\sum_{k=1}^{n+1}\frac{(n+1)!(k-1)!}{(n-k+1)!{\left(-a-n\right)_{k}}}z^{n-k+1}-% \sum_{k=0}^{\infty}\frac{{\left(a+n+1\right)_{k}}}{{\left(n+2\right)_{k}}k!}z^% {n+k+1}\left(\mathop{\ln\/}\nolimits z+\mathop{\psi\/}\nolimits\!\left(a+n+k+1% \right)-\mathop{\psi\/}\nolimits\!\left(1+k\right)-\mathop{\psi\/}\nolimits\!% \left(n+k+2\right)\right),$

if $a\neq 0,-1,-2,\dots$, or $z^{n+1}\mathop{M\/}\nolimits\!\left(a+n+1,n+2,z\right)$ and

 13.2.31 $\sum_{k=1}^{n+1}\frac{(n+1)!(k-1)!}{(n-k+1)!{\left(-a-n\right)_{k}}}z^{n-k+1}-% \sum_{k=0}^{-a-n-1}\frac{{\left(a+n+1\right)_{k}}}{{\left(n+2\right)_{k}}k!}z^% {n+k+1}\left(\mathop{\ln\/}\nolimits z+\mathop{\psi\/}\nolimits\!\left(-a-n-k% \right)-\mathop{\psi\/}\nolimits\!\left(1+k\right)-\mathop{\psi\/}\nolimits\!% \left(n+k+2\right)\right)+(-1)^{n-a}{(-a-n-1)!}\sum_{k=-a-n}^{\infty}\frac{(k+% a+n)!}{{\left(n+2\right)_{k}}k!}z^{n+k+1},$

if $a=-n-1,-n-2,-n-3,\dots$, or $z^{n+1}\mathop{M\/}\nolimits\!\left(a+n+1,n+2,z\right)$ and

 13.2.32 $\sum_{k=a+n+1}^{n+1}\frac{(k-1)!}{(n-k+1)!(k-a-n-1)!}z^{n-k+1},$ Symbols: $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable Referenced by: §13.2(v) Permalink: http://dlmf.nist.gov/13.2.E32 Encodings: TeX, pMML, png See also: Annotations for 13.2(v)

if $a=0,-1,\dots,-n$.

## §13.2(vi) Wronskians

 13.2.33 $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathbf{M}}\/}% \nolimits\!\left(a,b,z\right),z^{1-b}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a% -b+1,2-b,z\right)\right\}$ $\displaystyle=\mathop{\sin\/}\nolimits\!\left(\pi b\right)z^{-b}e^{z}/\pi,$ 13.2.34 $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathbf{M}}\/}% \nolimits\!\left(a,b,z\right),\mathop{U\/}\nolimits\!\left(a,b,z\right)\right\}$ $\displaystyle=-\ifrac{z^{-b}e^{z}}{\mathop{\Gamma\/}\nolimits\!\left(a\right)},$ 13.2.35 $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathbf{M}}\/}% \nolimits\!\left(a,b,z\right),e^{z}\mathop{U\/}\nolimits\!\left(b-a,b,e^{\pm% \pi\mathrm{i}}z\right)\right\}$ $\displaystyle=\ifrac{e^{\mp b\pi\mathrm{i}}z^{-b}e^{z}}{\mathop{\Gamma\/}% \nolimits\!\left(b-a\right)},$ 13.2.36 $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{z^{1-b}\mathop{{\mathbf{M}}% \/}\nolimits\!\left(a-b+1,2-b,z\right),\mathop{U\/}\nolimits\!\left(a,b,z% \right)\right\}$ $\displaystyle=-\ifrac{z^{-b}e^{z}}{\mathop{\Gamma\/}\nolimits\!\left(a-b+1% \right)},$ 13.2.37 $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{z^{1-b}\mathop{{\mathbf{M}}% \/}\nolimits\!\left(a-b+1,2-b,z\right),e^{z}\mathop{U\/}\nolimits\!\left(b-a,b% ,e^{\pm\pi\mathrm{i}}z\right)\right\}$ $\displaystyle=-\ifrac{z^{-b}e^{z}}{\mathop{\Gamma\/}\nolimits\!\left(1-a\right% )},$ 13.2.38 $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{U\/}\nolimits\!% \left(a,b,z\right),e^{z}\mathop{U\/}\nolimits\!\left(b-a,b,e^{\pm\pi\mathrm{i}% }z\right)\right\}$ $\displaystyle=e^{\pm(a-b)\pi\mathrm{i}}z^{-b}e^{z}.$

## §13.2(vii) Connection Formulas

### Kummer’s Transformations

 13.2.39 $\displaystyle\mathop{M\/}\nolimits\!\left(a,b,z\right)$ $\displaystyle=e^{z}\mathop{M\/}\nolimits\!\left(b-a,b,-z\right),$ Symbols: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of exponential function and $z$: complex variable A&S Ref: 13.1.27 Referenced by: §13.12, §13.8(i), §13.9(ii), §18.17(vi) Permalink: http://dlmf.nist.gov/13.2.E39 Encodings: TeX, pMML, png See also: Annotations for 13.2(vii) 13.2.40 $\displaystyle\mathop{U\/}\nolimits\!\left(a,b,z\right)$ $\displaystyle=z^{1-b}\mathop{U\/}\nolimits\!\left(a-b+1,2-b,z\right).$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.1.29 Referenced by: §13.8(iii) Permalink: http://dlmf.nist.gov/13.2.E40 Encodings: TeX, pMML, png See also: Annotations for 13.2(vii)
 13.2.41 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(b\right)}\mathop{M\/}\nolimits\!% \left(a,b,z\right)=\frac{e^{\mp a\pi\mathrm{i}}}{\mathop{\Gamma\/}\nolimits\!% \left(b-a\right)}\mathop{U\/}\nolimits\!\left(a,b,z\right)+\frac{e^{\pm(b-a)% \pi\mathrm{i}}}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}e^{z}\mathop{U\/}% \nolimits\!\left(b-a,b,e^{\pm\pi\mathrm{i}}z\right).$

Also, when $b$ is not an integer

 13.2.42 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=\frac{\mathop{\Gamma\/}\nolimits\!% \left(1-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)}\mathop{M\/}% \nolimits\!\left(a,b,z\right)+\frac{\mathop{\Gamma\/}\nolimits\!\left(b-1% \right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}z^{1-b}\mathop{M\/}% \nolimits\!\left(a-b+1,2-b,z\right).$