13 Confluent Hypergeometric Functions Kummer Functions 13.1 Special Notation 13.3 Recurrence Relations and Derivatives

§13.2 Definitions and Basic Properties

Keywords:: Kummer functions
Referenced by:: ¶ ‣ §33.22(ii), §9.10(v), §9.6(iii)
Permalink:: http://dlmf.nist.gov/13.2

§13.2(i) Differential Equation
§13.2(ii) Analytic Continuation
§13.2(iii) Limiting Forms as $z\to 0$
§13.2(iv) Limiting Forms as $z\to\infty$
§13.2(v) Numerically Satisfactory Solutions
§13.2(vi) Wronskians
§13.2(vii) Connection Formulas

§13.2(i) Differential Equation

Notes:: See Olver (1997b, Chapter 7, §§3, 9, 10) and Temme (1996b, §§7.1–7.2). (13.2.7) and (13.2.8) are terminating forms of the asymptotic expansion (13.7.3) (that the $\sim$ sign can be replaced by in these circumstances follows from (13.7.4) and (13.7.5).)
Referenced by:: ¶ ‣ §10.16, §10.22(iv), ¶ ‣ §10.39, ¶ ‣ §12.14(vii), §12.20, §12.7(iv), §13.6(vi), ¶ ‣ §18.11(i), §31.12, §33.14(ii), §33.2(ii), §33.2(iii), §6.11, §6.20(ii), ¶ ‣ §7.11, §7.18(iv), §8.19(vi), §8.5, Notation for the Special Functions
Permalink:: http://dlmf.nist.gov/13.2.i

¶ Kummer’s Equation

Keywords:: Kummer functions, Kummer’s equation

13.2.1 $z\frac{{d}^{2}w}{{dz}^{2}}+(b-z)\frac{dw}{dz}-aw=0.$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to and : 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

This equation has a regular singularity at the origin with indices 0 and , 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 by $\ifrac{z}{b}$ , letting $b\to\infty$ , and subsequently replacing the symbol by . In effect, the regular singularities of the hypergeometric differential equation at and $\infty$ coalesce into an irregular singularity at $\infty$ .

¶ Standard Solutions

Keywords:: Kummer functions, Kummer’s equation

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:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function
Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, : nonnegative integer and : 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

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:: $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, : nonnegative integer and : complex variable
Referenced by:: ¶ ‣ §13.2(i)
Permalink:: http://dlmf.nist.gov/13.2.E3
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function and : complex variable
Referenced by:: §13.2(ii)
Permalink:: http://dlmf.nist.gov/13.2.E4
Encodings:: TeX, pMML, png

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

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E5
Encodings:: TeX, pMML, png

When , $n=0,1,2,\dots$ , $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ is a polynomial in of degree not exceeding ; this is also true of $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ provided that 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:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function
Symbols:: $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, : 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

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

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

13.2.7 $\mathop{U\/}\nolimits\!\left(-n,b,z\right)=(-1)^{{n}}\sum_{{s=0}}^{{n}}\binom{% n}{s}\left(b+s\right)_{{n-s}}(-z)^{{s}}.$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\binom{m}{n}$ : binomial coefficient, : nonnegative integer, : nonnegative integer and : complex variable
Referenced by:: §13.2(i)
Permalink:: http://dlmf.nist.gov/13.2.E7
Encodings:: TeX, pMML, png

Similarly, when , $n=0,1,2,\ldots$ ,

13.2.8 $\mathop{U\/}\nolimits\!\left(a,a+n+1,z\right)=z^{{-a}}\sum_{{s=0}}^{{n}}\binom% {n}{s}\left(a\right)_{{s}}z^{{-s}}.$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\binom{m}{n}$ : binomial coefficient, : nonnegative integer, : nonnegative integer and : complex variable
Referenced by:: §13.2(i)
Permalink:: http://dlmf.nist.gov/13.2.E8
Encodings:: TeX, pMML, png

When , $n=0,1,2,\dots$ ,

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:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : nonnegative integer and : complex variable
Referenced by:: ¶ ‣ §13.2(i), §33.6, §6.6
Permalink:: http://dlmf.nist.gov/13.2.E9
Encodings:: TeX, pMML, png

if $a\neq 0,-1,-2,\dots$ , or

13.2.10 $\mathop{U\/}\nolimits\!\left(a,n+1,z\right)=(-1)^{a}\sum_{{k=0}}^{{-a}}\binom{% -a}{k}\left(n+k+1\right)_{{-a-k}}(-z)^{k},$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\binom{m}{n}$ : binomial coefficient, : nonnegative integer and : complex variable
Referenced by:: ¶ ‣ §13.2(i)
Permalink:: http://dlmf.nist.gov/13.2.E10
Encodings:: TeX, pMML, png

if $a=0,-1,-2,\dots$ .

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

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

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E11
Encodings:: TeX, pMML, png

§13.2(ii) Analytic Continuation

Notes:: To verify (13.2.12) replace the $\mathop{U\/}\nolimits$ functions by $\mathop{{\mathbf{M}}\/}\nolimits$ functions by means of (13.2.42) and (13.2.4), then recall that each $\mathop{{\mathbf{M}}\/}\nolimits$ function is an entire function of .
Keywords:: Kummer functions, analytic continuation
Permalink:: http://dlmf.nist.gov/13.2.ii

When $m\in\Integer$ ,

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function, : integer and : complex variable
A&S Ref:: 13.1.10 (in different form)
Referenced by:: §13.2(ii)
Permalink:: http://dlmf.nist.gov/13.2.E12
Encodings:: TeX, pMML, png

Except when each branch of $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ is entire in and . 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 $z\to 0$

Notes:: See Temme (1996b, Ex. 7.6: an error in the equation that corresponds to (13.2.19) has been corrected).
Keywords:: Kummer functions
Referenced by:: §13.14(iii)
Permalink:: http://dlmf.nist.gov/13.2.iii

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E13
Encodings:: TeX, pMML, png

Next, in cases when or , where is a nonnegative integer,

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E14
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E15
Encodings:: TeX, pMML, png

In all other cases

13.2.16 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=\frac{\mathop{\Gamma\/}\nolimits\!% \left(b-1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}z^{{1-b}}+\mathop% {O\/}\nolimits\!\left(z^{{2-\realpart{b}}}\right),$ $\realpart{b}\geq 2$ , $b\not=2$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\realpart{}$ : real part and : complex variable
A&S Ref:: 13.5.6 (with order estimate corrected)
Permalink:: http://dlmf.nist.gov/13.2.E16
Encodings:: TeX, pMML, png

13.2.17 $\mathop{U\/}\nolimits\!\left(a,2,z\right)=\frac{1}{\mathop{\Gamma\/}\nolimits% \!\left(a\right)}z^{{-1}}+\mathop{O\/}\nolimits\!\left(\mathop{\ln\/}\nolimits z% \right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : complex variable
A&S Ref:: 13.5.7
Permalink:: http://dlmf.nist.gov/13.2.E17
Encodings:: TeX, pMML, png

13.2.18 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=\frac{\mathop{\Gamma\/}\nolimits\!% \left(b-1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}z^{{1-b}}+\frac{% \mathop{\Gamma\/}\nolimits\!\left(1-b\right)}{\mathop{\Gamma\/}\nolimits\!% \left(a-b+1\right)}+\mathop{O\/}\nolimits\!\left(z^{{2-\realpart{b}}}\right),$ $1\leq\realpart{b}<2$ , $b\not=1$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\realpart{}$ : real part and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E18
Encodings:: TeX, pMML, png

13.2.19 $\mathop{U\/}\nolimits\!\left(a,1,z\right)=-\frac{1}{\mathop{\Gamma\/}\nolimits% \!\left(a\right)}\left(\mathop{\ln\/}\nolimits z+\mathop{\psi\/}\nolimits\!% \left(a\right)+2\gamma\right)+\mathop{O\/}\nolimits\!\left(z\mathop{\ln\/}% \nolimits z\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\EulerConstant$ : Euler’s constant, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : complex variable
A&S Ref:: 13.5.9 (as corrected in later editions)
Referenced by:: §13.2(iii)
Permalink:: http://dlmf.nist.gov/13.2.E19
Encodings:: TeX, pMML, png

13.2.20 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=\frac{\mathop{\Gamma\/}\nolimits\!% \left(1-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)}+\mathop{O\/}% \nolimits\!\left(z^{{1-\realpart{b}}}\right),$ $0<\realpart{b}<1$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\realpart{}$ : real part and : complex variable
A&S Ref:: 13.5.10
Permalink:: http://dlmf.nist.gov/13.2.E20
Encodings:: TeX, pMML, png

13.2.21 $\mathop{U\/}\nolimits\!\left(a,0,z\right)=\frac{1}{\mathop{\Gamma\/}\nolimits% \!\left(a+1\right)}+\mathop{O\/}\nolimits\!\left(z\mathop{\ln\/}\nolimits z% \right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : complex variable
A&S Ref:: 13.5.11
Permalink:: http://dlmf.nist.gov/13.2.E21
Encodings:: TeX, pMML, png

13.2.22 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=\frac{\mathop{\Gamma\/}\nolimits\!% \left(1-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)}+\mathop{O\/}% \nolimits\!\left(z\right),$ $\realpart{b}\leq 0$ , $b\not=0$ .

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\realpart{}$ : real part and : complex variable
A&S Ref:: 13.5.12
Permalink:: http://dlmf.nist.gov/13.2.E22
Encodings:: TeX, pMML, png

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

Notes:: See (13.7.2).
Keywords:: Kummer functions
Permalink:: http://dlmf.nist.gov/13.2.iv

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, : complex variable and $\delta$ : small positive constant
Permalink:: http://dlmf.nist.gov/13.2.E23
Encodings:: TeX, pMML, png

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

Notes:: See Olver (1997b, Chapter 7, §§9, 10). (13.2.27)–(13.2.32) are obtained by considering limiting forms of the connection formulas in §13.2(vii); see Olver (1997b, Chapter 7, Ex. 10.6) and Slater (1960, §§1.5–1.5.1).
Keywords:: Kummer’s equation
Referenced by:: §13.14(v)
Permalink:: http://dlmf.nist.gov/13.2.v

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

13.2.24

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

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

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E24
Encodings:: TeX, TeX, pMML, pMML, png, png

13.2.25

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

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

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E25
Encodings:: TeX, TeX, pMML, pMML, png, png

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

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

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\Integer$ : set of all integers, $\notin$ : not an element of and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E26
Encodings:: TeX, pMML, png

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : nonnegative integer and : complex variable
Referenced by:: §13.2(v)
Permalink:: http://dlmf.nist.gov/13.2.E27
Encodings:: TeX, pMML, png

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

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E28
Encodings:: TeX, pMML, png

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, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E29
Encodings:: TeX, pMML, png

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E30
Encodings:: TeX, pMML, png

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E31
Encodings:: TeX, pMML, png

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, : nonnegative integer and : complex variable
Referenced by:: §13.2(v)
Permalink:: http://dlmf.nist.gov/13.2.E32
Encodings:: TeX, pMML, png

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

§13.2(vi) Wronskians

Notes:: See Slater (1960, §2.1.2).
Keywords:: Kummer functions
Permalink:: http://dlmf.nist.gov/13.2.vi

13.2.33 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathbf{M}}\/}\nolimits\!\left(% a,b,z\right),z^{{1-b}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a-b+1,2-b,z% \right)\right\}=\mathop{\sin\/}\nolimits\!\left(\pi b\right)z^{{-b}}e^{z}/\pi,$

Symbols:: $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function and : complex variable
A&S Ref:: 13.1.20
Permalink:: http://dlmf.nist.gov/13.2.E33
Encodings:: TeX, pMML, png

13.2.34 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathbf{M}}\/}\nolimits\!\left(% a,b,z\right),\mathop{U\/}\nolimits\!\left(a,b,z\right)\right\}=-\ifrac{z^{{-b}% }e^{z}}{\mathop{\Gamma\/}\nolimits\!\left(a\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : base of exponential function and : complex variable
A&S Ref:: 13.1.22
Permalink:: http://dlmf.nist.gov/13.2.E34
Encodings:: TeX, pMML, png

13.2.35 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathbf{M}}\/}\nolimits\!\left(% a,b,z\right),e^{z}\mathop{U\/}\nolimits\!\left(b-a,b,e^{{\pm\pi i}}z\right)% \right\}=\ifrac{e^{{\mp b\pi i}}z^{{-b}}e^{z}}{\mathop{\Gamma\/}\nolimits\!% \left(b-a\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : base of exponential function and : complex variable
A&S Ref:: 13.1.23
Permalink:: http://dlmf.nist.gov/13.2.E35
Encodings:: TeX, pMML, png

13.2.36 $\mathop{\mathscr{W}\/}\nolimits\left\{z^{{1-b}}\mathop{{\mathbf{M}}\/}% \nolimits\!\left(a-b+1,2-b,z\right),\mathop{U\/}\nolimits\!\left(a,b,z\right)% \right\}=-\ifrac{z^{{-b}}e^{z}}{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : base of exponential function and : complex variable
A&S Ref:: 13.1.24
Permalink:: http://dlmf.nist.gov/13.2.E36
Encodings:: TeX, pMML, png

13.2.37 $\mathop{\mathscr{W}\/}\nolimits\left\{z^{{1-b}}\mathop{{\mathbf{M}}\/}% \nolimits\!\left(a-b+1,2-b,z\right),e^{z}\mathop{U\/}\nolimits\!\left(b-a,b,e^% {{\pm\pi i}}z\right)\right\}=-\ifrac{z^{{-b}}e^{z}}{\mathop{\Gamma\/}\nolimits% \!\left(1-a\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : base of exponential function and : complex variable
A&S Ref:: 13.1.25
Permalink:: http://dlmf.nist.gov/13.2.E37
Encodings:: TeX, pMML, png

13.2.38 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{U\/}\nolimits\!\left(a,b,z\right% ),e^{z}\mathop{U\/}\nolimits\!\left(b-a,b,e^{{\pm\pi i}}z\right)\right\}=e^{{% \pm(a-b)\pi i}}z^{{-b}}e^{z}.$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : base of exponential function and : complex variable
A&S Ref:: 13.1.26
Permalink:: http://dlmf.nist.gov/13.2.E38
Encodings:: TeX, pMML, png

§13.2(vii) Connection Formulas

Notes:: See Temme (1996b, §§7.1, 7.2).
Keywords:: Kummer functions
Referenced by:: §13.2(v), §13.29(ii), §33.14(ii), §33.2(ii), §8.5
Permalink:: http://dlmf.nist.gov/13.2.vii

¶ Kummer’s Transformations

Keywords:: Kummer functions, Kummer’s transformations

13.2.39 $\mathop{M\/}\nolimits\!\left(a,b,z\right)=e^{z}\mathop{M\/}\nolimits\!\left(b-% a,b,-z\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function and : 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

13.2.40 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=z^{{1-b}}\mathop{U\/}\nolimits\!% \left(a-b+1,2-b,z\right).$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function and : 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

13.2.41 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(b\right)}\mathop{M\/}\nolimits\!% \left(a,b,z\right)=\frac{e^{{\mp a\pi i}}}{\mathop{\Gamma\/}\nolimits\!\left(b% -a\right)}\mathop{U\/}\nolimits\!\left(a,b,z\right)+\frac{e^{{\pm(b-a)\pi i}}}% {\mathop{\Gamma\/}\nolimits\!\left(a\right)}e^{z}\mathop{U\/}\nolimits\!\left(% b-a,b,e^{{\pm\pi i}}z\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function and : complex variable
Referenced by:: §13.12, §13.14(vii), §13.4(i), §13.7(ii)
Permalink:: http://dlmf.nist.gov/13.2.E41
Encodings:: TeX, pMML, png

Also, when is not an integer

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function and : complex variable
A&S Ref:: 13.1.3 (in different form)
Referenced by:: §13.14(vii), §13.2(ii), §13.6(v), §13.9(ii), §33.13
Permalink:: http://dlmf.nist.gov/13.2.E42
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 13.1 Special Notation 13.3 Recurrence Relations and Derivatives