Kummer equation

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1: 13.2 Definitions and Basic Properties

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Kummer’s Equation

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

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Symbols:

$\frac{df}{dx}$: 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:

pMML, png, TeX

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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Standard Solutions

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§13.2(v) Numerically Satisfactory Solutions

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2: 13.3 Recurrence Relations and Derivatives

… ►Kummer’s differential equation (13.2.1) is equivalent to … ►

13.3.14 $$(a+1)zU(a+2,b+2,z)+(z-b)U(a+1,b+1,z)-U(a,b,z)=0.$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.3(i)

Permalink:

http://dlmf.nist.gov/13.3.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

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3: 13.14 Definitions and Basic Properties

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Whittaker’s Equation

… ►This equation is obtained from Kummer’s equation (13.2.1) via the substitutions $W={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}+\mu }w$, $\kappa =\frac{1}{2}b-a$, and $\mu =\frac{1}{2}b-\frac{1}{2}$. … ►Standard solutions are: … ►The principal branches correspond to the principal branches of the functions $z^{\frac{1}{2}+\mu }$ and $U(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z)$ on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i). …

4: 15.10 Hypergeometric Differential Equation

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§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

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5: 13.29 Methods of Computation

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§13.29(ii) Differential Equations

►A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods. … ►The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. In Allasia and Besenghi (1991) and Allasia and Besenghi (1987a) the high accuracy of the trapezoidal rule for the computation of Kummer functions is described. … ►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of $M(n,b,x)$, when $b$ and $x$ are real and $n$ is a positive integer. …

6: Errata

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There were clarifications made in the conditions on the parameter $a$ in $U(a,b,z)$ of those equations.

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A new paragraph with several new equations and a new reference has been added at the end to provide asymptotic expansions for Kummer functions $U(a,b,z)$ and $𝐌(a,b,z)$ as $a\to \mathrm{\infty }$ in $|\mathrm{ph}a|\le \pi -\delta$ and $b$ and $z$ fixed.

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13.2.7 $$U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)={(-1)}^m\sum _{s=0}^m\left(\genfrac{}{}{0.0pt}{}{m}{s}\right){\left(b+s\right)}_{m-s}{(-z)}^s$$

The equality $U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)$ 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 by Adri Olde Daalhuis.

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13.2.8 $$U(a,a+n+1,z)=\frac{{(-1)}^n{\left(1-a-n\right)}_n}{z^{a+n}}M(-n,1-a-n,z)=z^{-a}\sum _{s=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{s}\right){\left(a\right)}_s{z}^{-s}$$

The equality $U(a,a+n+1,z)=\frac{{(-1)}^n{\left(1-a-n\right)}_n}{z^{a+n}}M(-n,1-a-n,z)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation.

Reported 2015-02-10 by Adri Olde Daalhuis.

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13.2.10 $$U(-m,n+1,z)={(-1)}^m{\left(n+1\right)}_{m}M(-m,n+1,z)={(-1)}^m\sum _{s=0}^m\left(\genfrac{}{}{0.0pt}{}{m}{s}\right){\left(n+s+1\right)}_{m-s}{(-z)}^s$$

The equality $U(-m,n+1,z)={(-1)}^m{\left(n+1\right)}_{m}M(-m,n+1,z)$ 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 $a=-m,m=0,1,2,\mathrm{\dots }$ has been introduced.

Reported 2015-02-10 by Adri Olde Daalhuis.

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7: 31.11 Expansions in Series of Hypergeometric Functions

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31.11.3_1 $$P_j^5=\frac{{\left(\lambda \right)}_j{\left(1-\gamma +\lambda \right)}_j}{{\left(1+\lambda -\mu \right)}_{2j}}{z}^{-\lambda -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\lambda +j,1-\gamma +\lambda +j}{1+\lambda -\mu +2j};\frac{1}{z}),$$

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Defines:

$P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.3_2 $$P_j^6=\frac{{\left(\lambda -\mu \right)}_{2j}}{{\left(1-\mu \right)}_j{\left(\gamma -\mu \right)}_j}{z}^{-\mu +j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\mu -j,1-\gamma +\mu -j}{1-\lambda +\mu -2j};\frac{1}{z}).$$

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Defines:

$P_j^6$: second Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.12 $$P_j^5=\frac{{\left(\alpha \right)}_j{\left(1-\gamma +\alpha \right)}_j}{{\left(1+\alpha -\beta +ϵ\right)}_{2j}}{z}^{-\alpha -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +j,1-\gamma +\alpha +j}{1+\alpha -\beta +ϵ+2j};\frac{1}{z}),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $\alpha$: real or complex parameter, $\beta$: real or complex parameter and $P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation

Referenced by:

§31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E12

Encodings:

pMML, png, TeX

Notational Change (effective with 1.1.7):

We have rewritten the gamma functions in the prefactor more concisely using Pochhammer symbols.

See also:

Annotations for §31.11(iii), §31.11 and Ch.31

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8: 13.28 Physical Applications

§13.28 Physical Applications

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§13.28(i) Exact Solutions of the Wave Equation

►The reduced wave equation ${\nabla }^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi \eta }\cos \varphi$, $y=2\sqrt{\xi \eta }\sin \varphi$, $z=\xi -\eta$, can be solved via separation of variables $w=f_1(\xi )f_2(\eta ){\mathrm{e}}^{\mathrm{i}p\varphi }$, where …and $V_{\kappa ,\mu }^{(j)}(z)$, $j=1,2$, denotes any pair of solutions of Whittaker’s equation (13.14.1). …

9: 31.3 Basic Solutions

… ►The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).

10: 13.8 Asymptotic Approximations for Large Parameters

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13.8.11 $$U(a,b,z)\sim 2{\left(z/a\right)}^{(1-b)/2}\frac{{\mathrm{e}}^{z/2}}{\mathrm{\Gamma }\left(a\right)}\left(K_{b-1}\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{a^s}+\sqrt{z/a}{K}_b\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{a^s}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.8(iii)

Permalink:

http://dlmf.nist.gov/13.8.E11

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added.

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

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13.8.12 $$𝐌(a,b,z)\sim {\left(z/a\right)}^{(1-b)/2}\frac{{\mathrm{e}}^{z/2}\mathrm{\Gamma }\left(1+a-b\right)}{\mathrm{\Gamma }\left(a\right)}\left(I_{b-1}\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{a^s}-\sqrt{z/a}{I}_b\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{a^s}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.8.E12

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added.

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

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13.8.13 $$𝐌(-a,b,z)\sim {\left(z/a\right)}^{(1-b)/2}\frac{{\mathrm{e}}^{z/2}\mathrm{\Gamma }\left(1+a\right)}{\mathrm{\Gamma }\left(a+b\right)}\left(J_{b-1}\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{{(-a)}^s}-\sqrt{z/a}{J}_b\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{{(-a)}^s}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.8.13)

Permalink:

http://dlmf.nist.gov/13.8.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added. Expansion (13.8.13) is (10.3.58) in Temme (2015). In that reference the ${(-1)}^s$ is missing.

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

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13.8.14 $$U(-a,b,z)\sim {\left(z/a\right)}^{(1-b)/2}{\mathrm{e}}^{z/2}\mathrm{\Gamma }\left(1+a\right)\left(C_{b-1}(a,2\sqrt{az})\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{{(-a)}^s}-\sqrt{z/a}{C}_b(a,2\sqrt{az})\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{{(-a)}^s}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $(a,b)$: open interval, $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.8.14)

Permalink:

http://dlmf.nist.gov/13.8.E14

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added. Expansion (13.8.14) is (10.3.68) in Temme (2015). In that reference the ${(-1)}^s$ is missing.

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

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