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

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13.3.4 $$bM(a,b,z)-bM(a-1,b,z)-zM(a,b+1,z)=0,$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.4

Permalink:

http://dlmf.nist.gov/13.3.E4

Encodings:

pMML, png, TeX

See also:

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

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13.3.7 $$U(a-1,b,z)+(b-2a-z)U(a,b,z)+a(a-b+1)U(a+1,b,z)=0,$$

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Symbols:

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

A&S Ref:

13.4.15

Permalink:

http://dlmf.nist.gov/13.3.E7

Encodings:

pMML, png, TeX

See also:

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

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13.3.9 $$U(a,b,z)-aU(a+1,b,z)-U(a,b-1,z)=0,$$

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Symbols:

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

A&S Ref:

13.4.17

Permalink:

http://dlmf.nist.gov/13.3.E9

Encodings:

pMML, png, TeX

See also:

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

… ►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: 13.6 Relations to Other Functions

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13.6.1 $$M(a,a,z)={\mathrm{e}}^z,$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

13.6.12

Referenced by:

§13.10(v)

Permalink:

http://dlmf.nist.gov/13.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(i), §13.6 and Ch.13

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13.6.2 $$M(1,2,2z)=\frac{{\mathrm{e}}^z}{z}\sinh z,$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

13.6.14

Permalink:

http://dlmf.nist.gov/13.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(i), §13.6 and Ch.13

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13.6.3 $$M(0,b,z)=U(0,b,z)=1,$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(i), §13.6 and Ch.13

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13.6.4 $$U(a,a+1,z)=z^{-a}.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/13.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(i), §13.6 and Ch.13

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13.6.18 $$U(\frac{1}{2}-\frac{1}{2}n,\frac{3}{2},z^2)=2^{-n}{z}^{-1}{H}_n\left(z\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $U(a,b,z)$: Kummer confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.6.38 (as corrected in later editions)

Referenced by:

§13.6(v)

Permalink:

http://dlmf.nist.gov/13.6.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

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4: 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: … ►

13.14.5 $$U(a,b,z)={\mathrm{e}}^{\frac{1}{2}z}{z}^{-\frac{1}{2}b}{W}_{\frac{1}{2}b-a,\frac{1}{2}b-\frac{1}{2}}\left(z\right).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

§10.16, §13.11, §13.14(iii), §13.14(vii), §13.16(i), §13.18(i), §13.18(ii), §13.18(iii), §13.18(iv), §13.18(v), §13.23(iii), §13.8(ii), §13.8(ii), §13.8(iii)

Permalink:

http://dlmf.nist.gov/13.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

… ►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). …

5: 13.12 Products

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13.12.1 $$M(a,b,z)M(-a,-b,-z)+\frac{a(a-b)z^2}{b^2(1-b^2)}M(1+a,2+b,z)M(1-a,2-b,-z)=1.$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.12, §13.25

Permalink:

http://dlmf.nist.gov/13.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.12 and Ch.13

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6: 13.4 Integral Representations

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13.4.1 $$𝐌(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b-a\right)}{\int }_0^1{\mathrm{e}}^{zt}{t}^{a-1}{(1-t)}^{b-a-1}dt,$$ $\mathrm{\Re }b>\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

13.2.1 (in different form)

Referenced by:

§13.10(v), §13.29(iii)

Permalink:

http://dlmf.nist.gov/13.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

►

13.4.2 $$𝐌(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(b-c\right)}{\int }_0^1𝐌(a,c,zt)t^{c-1}{(1-t)}^{b-c-1}dt,$$ $\mathrm{\Re }b>\mathrm{\Re }c>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $c$: arbitrary

Referenced by:

§13.10(vi)

Permalink:

http://dlmf.nist.gov/13.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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13.4.3 $$𝐌(a,b,-z)=\frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\mathrm{\Gamma }\left(a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{a-\frac{1}{2}b-\frac{1}{2}}{J}_{b-1}\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }a>0$.

ⓘ

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, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.10(v)

Permalink:

http://dlmf.nist.gov/13.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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13.4.4 $$U(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{a-1}{(1+t)}^{b-a-1}dt,$$ $\mathrm{\Re }a>0$, ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

13.2.5

Referenced by:

§13.10(v), §13.29(iii), §7.11, (9.5.8)

Permalink:

http://dlmf.nist.gov/13.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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13.4.13 $$𝐌(a,b,z)=\frac{z^{1-b}}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+,1+)}{\mathrm{e}}^{zt}{t}^{-b}{\left(1-\frac{1}{t}\right)}^{-a}dt,$$ .

ⓘ

Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $(a,b)$: open interval, $\mathrm{ph}$: phase and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(ii), §13.4 and Ch.13

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7: 13.11 Series

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13.11.1 $$M(a,b,z)=\mathrm{\Gamma }\left(a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{\frac{1}{2}-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(2a-1\right)}_s{\left(2a-b\right)}_s}{{\left(b\right)}_{s}s!}\left(a-\frac{1}{2}+s\right)I_{a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.6(iii), §13.6(iii), §8.7

Permalink:

http://dlmf.nist.gov/13.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.11 and Ch.13

►

13.11.2 $$M(a,b,z)=\mathrm{\Gamma }\left(b-a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{a-b+\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(2b-2a-1\right)}_s{\left(b-2a\right)}_s(b-a-\frac{1}{2}+s)}{{\left(b\right)}_{s}s!}{I}_{b-a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $b-a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Proof sketch:

Combine (13.24.1) with (13.14.4).

A&S Ref:

13.3.6

Referenced by:

§13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

… ►

13.11.3 $$𝐌(a,b,z)={\mathrm{e}}^{\frac{1}{2}z}\sum _{s=0}^{\mathrm{\infty }}{A}_s{\left(b-2a\right)}^{\frac{1}{2}(1-b-s)}{\left(\frac{1}{2}z\right)}^{\frac{1}{2}(1-b+s)}{J}_{b-1+s}\left(\sqrt{2z(b-2a)}\right),$$

ⓘ

Symbols:

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

Source:

Slater (1960, §3.8)

A&S Ref:

13.3.7

Referenced by:

§13.11, Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

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8: 13.8 Asymptotic Approximations for Large Parameters

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13.8.5 $$U(a,b,z)\sim b^{-\frac{1}{2}a}{\mathrm{e}}^{\frac{1}{4}{\zeta }^{2}b}\left(\lambda {\left(\frac{\lambda -1}{\zeta }\right)}^{a-1}U(a-\frac{1}{2},\zeta \sqrt{b})-\left(\lambda {\left(\frac{\lambda -1}{\zeta }\right)}^{a-1}-{\left(\frac{\zeta }{\lambda -1}\right)}^a\right)\frac{U(a-\frac{3}{2},\zeta \sqrt{b})}{\zeta \sqrt{b}}\right)$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.8.E5

Encodings:

pMML, png, TeX

See also:

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

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13.8.6 $$M(a,b,b)=\sqrt{\pi }{\left(\frac{b}{2}\right)}^{\frac{1}{2}a}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{2}(a+1)\right)}+\frac{(a+1)\sqrt{8/b}}{3\mathrm{\Gamma }\left(\frac{1}{2}a\right)}+O\left(\frac{1}{b}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/13.8.E6

Encodings:

pMML, png, TeX

See also:

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

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13.8.7 $$U(a,b,b)=\sqrt{\pi }{\left(2b\right)}^{-\frac{1}{2}a}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{2}(a+1)\right)}-\frac{(a+1)\sqrt{8/b}}{3\mathrm{\Gamma }\left(\frac{1}{2}a\right)}+O\left(\frac{1}{b}\right)\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/13.8.E7

Encodings:

pMML, png, TeX

See also:

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

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13.8.17 $$M(a,b,z)={\mathrm{e}}^{\nu z}\frac{{\mathrm{\Gamma }}^{\ast }(b)}{{\mathrm{\Gamma }}^{\ast }(a)}\left(1+\frac{(1-\nu )(1+6{\nu }^2{z}^2)}{12a}+O\left(\frac{1}{\min (a^2,b^2)}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.8.E17

Encodings:

pMML, png, TeX

See also:

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

►

13.8.18 $$U(a,b+1,z)=z^{-b}{\mathrm{e}}^{(1-\nu )z}\frac{\mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }\left(a\right)}\left(1+\frac{\nu z(1-\nu )(2-\nu z)}{2a}+O\left(\frac{1}{\min (a^2,b^2)}\right)\right),$$ $\mathrm{\Re }z>0$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.8.E18

Encodings:

pMML, png, TeX

See also:

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

…

9: 15.10 Hypergeometric Differential Equation

… ►

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

… ►

15.10.17 $$w_3(z)=\frac{\mathrm{\Gamma }\left(1-c\right)\mathrm{\Gamma }\left(a+b-c+1\right)}{\mathrm{\Gamma }\left(a-c+1\right)\mathrm{\Gamma }\left(b-c+1\right)}{w}_1(z)+\frac{\mathrm{\Gamma }\left(c-1\right)\mathrm{\Gamma }\left(a+b-c+1\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{w}_2(z),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_2(z)$: Kummer’s solution $w_2$ and $w_3(z)$: Kummer’s solution $w_3$

Permalink:

http://dlmf.nist.gov/15.10.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

►

15.10.18 $$w_4(z)=\frac{\mathrm{\Gamma }\left(1-c\right)\mathrm{\Gamma }\left(c-a-b+1\right)}{\mathrm{\Gamma }\left(1-a\right)\mathrm{\Gamma }\left(1-b\right)}{w}_1(z)+\frac{\mathrm{\Gamma }\left(c-1\right)\mathrm{\Gamma }\left(c-a-b+1\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_2(z),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_2(z)$: Kummer’s solution $w_2$ and $w_4(z)$: Kummer’s solution $w_4$

Permalink:

http://dlmf.nist.gov/15.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

… ►

15.10.21 $$w_1(z)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(c-a-b\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_3(z)+\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a+b-c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{w}_4(z),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_3(z)$: Kummer’s solution $w_3$ and $w_4(z)$: Kummer’s solution $w_4$

Referenced by:

§15.8(i)

Permalink:

http://dlmf.nist.gov/15.10.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.25 $$w_1(z)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(b-a\right)}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)}{w}_5(z)+\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a-b\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_6(z),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_5(z)$: Kummer’s solution $w_5$ and $w_6(z)$: Kummer’s solution $w_6$

Referenced by:

§15.8(i)

Permalink:

http://dlmf.nist.gov/15.10.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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10: 13.29 Methods of Computation

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13.29.6 $$w(n)={\left(a\right)}_{n}U(n+a,b,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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