shift of variable

SearchAdvancedHelp

(0.003 seconds)

21—30 of 135 matching pages

21: 18.5 Explicit Representations

… ►

18.5.7 $$P_n^{(\alpha ,\beta )}\left(x\right)=\sum _{\mathrm{\ell }=0}^n\frac{{\left(n+\alpha +\beta +1\right)}_{\mathrm{\ell }}{\left(\alpha +\mathrm{\ell }+1\right)}_{n-\mathrm{\ell }}}{\mathrm{\ell }!(n-\mathrm{\ell })!}{\left(\frac{x-1}{2}\right)}^{\mathrm{\ell }}=\frac{{\left(\alpha +1\right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +\beta +1}{\alpha +1};\frac{1-x}{2}),$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.42

Referenced by:

§18.17(iv), §18.17(vii), (18.27.12_5), Table 18.3.1, §18.5(iii), §18.5(iii), §18.6(ii), §18.9(iii)

Permalink:

http://dlmf.nist.gov/18.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

►

18.5.8 $$P_n^{(\alpha ,\beta )}\left(x\right)=2^{-n}\sum _{\mathrm{\ell }=0}^n\left(\genfrac{}{}{0.0pt}{}{n+\alpha }{\mathrm{\ell }}\right)\left(\genfrac{}{}{0.0pt}{}{n+\beta }{n-\mathrm{\ell }}\right){(x-1)}^{n-\mathrm{\ell }}{(x+1)}^{\mathrm{\ell }}=\frac{{\left(\alpha +1\right)}_n}{n!}{\left(\frac{x+1}{2}\right)}^n{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,-n-\beta }{\alpha +1};\frac{x-1}{x+1}),$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.3.1

Referenced by:

§18.5(iii), §18.5(iii)

Permalink:

http://dlmf.nist.gov/18.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►

18.5.9 $$C_n^{(\lambda )}\left(x\right)=\frac{{\left(2\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,n+2\lambda }{\lambda +\frac{1}{2}};\frac{1-x}{2}),$$

ⓘ

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), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.46

Referenced by:

§18.5(iii), §18.6(i)

Permalink:

http://dlmf.nist.gov/18.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

►

18.5.10 $$C_n^{(\lambda )}\left(x\right)=\sum _{\mathrm{\ell }=0}^{\lfloor n/2\rfloor }\frac{{(-1)}^{\mathrm{\ell }}{\left(\lambda \right)}_{n-\mathrm{\ell }}}{\mathrm{\ell }!(n-2\mathrm{\ell })!}{(2x)}^{n-2\mathrm{\ell }}={(2x)}^n\frac{{\left(\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}n,-\frac{1}{2}n+\frac{1}{2}}{1-\lambda -n};\frac{1}{x^2}),$$

ⓘ

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), $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.3.4

Referenced by:

§18.17(vii), §18.5(iii), §18.6(ii)

Permalink:

http://dlmf.nist.gov/18.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►

18.5.12 $$L_n^{(\alpha )}\left(x\right)=\sum _{\mathrm{\ell }=0}^n\frac{{\left(\alpha +\mathrm{\ell }+1\right)}_{n-\mathrm{\ell }}}{(n-\mathrm{\ell })!\mathrm{\ell }!}{(-x)}^{\mathrm{\ell }}=\frac{{\left(\alpha +1\right)}_n}{n!}{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n}{\alpha +1};x).$$

ⓘ

Symbols:

${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.54

Referenced by:

§18.11(i), §18.17(v), §18.17(vi), §18.17(vii), §18.18(iii), §18.21(ii), Table 18.3.1, (18.34.1), §18.34(i), §18.5(iii), §18.6(ii)

Permalink:

http://dlmf.nist.gov/18.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

…

22: 16.12 Products

… ►

16.12.3 $${\left({}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};z)\right)}^2=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(2a\right)}_k{\left(2b\right)}_k{\left(c-\frac{1}{2}\right)}_k}{{\left(c\right)}_k{\left(2c-1\right)}_{k}k!}{}_4{}^{}F_3^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}}{a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-c};1)z^k,$$ .

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.12

Permalink:

http://dlmf.nist.gov/16.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.12 and Ch.16

…

23: 18.23 Hahn Class: Generating Functions

… ►

18.23.1 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-x}{\alpha +1};-z){}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{x-N}{\beta +1};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n}{{\left(\beta +1\right)}_{n}n!}{Q}_n(x;\alpha ,\beta ,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $N$: positive integer, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.23

Permalink:

http://dlmf.nist.gov/18.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

►

18.23.2 $${}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-x,-x+\beta +N+1}{-};-z){}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{x-N,x+\alpha +1}{-};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n{\left(\alpha +1\right)}_n}{n!}{Q}_n(x;\alpha ,\beta ,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $N$: positive integer, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.23

Permalink:

http://dlmf.nist.gov/18.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

… ►

18.23.4 $${\left(1-\frac{z}{c}\right)}^x{(1-z)}^{-x-\beta }=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\beta \right)}_n}{n!}{M}_n(x;\beta ,c)z^n,$$ $x=0,1,2,\mathrm{\dots }$, .

ⓘ

Symbols:

$M_n(x;\beta ,c)$: Meixner polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii)

Permalink:

http://dlmf.nist.gov/18.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

… ►

18.23.6 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a+\mathrm{i}x}{2\mathrm{\Re }a};-\mathrm{i}z){}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\overline{b}-\mathrm{i}x}{2\mathrm{\Re }b};\mathrm{i}z)=\sum _{n=0}^{\mathrm{\infty }}\frac{p_n(x;a,b,\overline{a},\overline{b})}{{\left(2\mathrm{\Re }a\right)}_n{\left(2\mathrm{\Re }b\right)}_n}{z}^n.$$

ⓘ

Symbols:

${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.23

Permalink:

http://dlmf.nist.gov/18.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

…

24: 18.7 Interrelations and Limit Relations

… ►

18.7.1 $$C_n^{(\lambda )}\left(x\right)=\frac{{\left(2\lambda \right)}_n}{{\left(\lambda +\frac{1}{2}\right)}_n}{P}_n^{(\lambda -\frac{1}{2},\lambda -\frac{1}{2})}\left(x\right),$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.27

Referenced by:

§18.15(ii), §18.7(i), §18.7(ii), §18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

►

18.7.2 $$P_n^{(\alpha ,\alpha )}\left(x\right)=\frac{{\left(\alpha +1\right)}_n}{{\left(2\alpha +1\right)}_n}{C}_n^{(\alpha +\frac{1}{2})}\left(x\right).$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.20

Referenced by:

§18.6(i), §18.7(i)

Permalink:

http://dlmf.nist.gov/18.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

… ►

18.7.7 $$T_n^{\ast }\left(x\right)=T_n\left(2x-1\right),$$

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $T_n^{\ast }\left(x\right)$: shifted Chebyshev polynomial of the first kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.14

Referenced by:

§18.7(i)

Permalink:

http://dlmf.nist.gov/18.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

►

18.7.8 $$U_n^{\ast }\left(x\right)=U_n\left(2x-1\right).$$

ⓘ

Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $U_n^{\ast }\left(x\right)$: shifted Chebyshev polynomial of the second kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.15

Permalink:

http://dlmf.nist.gov/18.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

… ►

18.7.10 $$P_n^{\ast }\left(x\right)=P_n\left(2x-1\right).$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $P_n^{\ast }\left(x\right)$: shifted Legendre polynomial, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

…

25: 6.2 Definitions and Interrelations

… ►

6.2.10 $$\mathrm{si}\left(z\right)=-{\int }_z^{\mathrm{\infty }}\frac{\sin t}{t}dt=\mathrm{Si}\left(z\right)-\frac{1}{2}\pi .$$

ⓘ

Defines:

$\mathrm{si}\left(z\right)$: sine integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral and $z$: complex variable

A&S Ref:

5.2.5 (in modified form) 5.2.26 (in modified form)

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(ii), §6.2 and Ch.6

… ►

6.2.17 $$\mathrm{f}\left(z\right)=\mathrm{Ci}\left(z\right)\sin z-\mathrm{si}\left(z\right)\cos z,$$

ⓘ

Defines:

$\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals

Symbols:

$\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{si}\left(z\right)$: sine integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.6

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.2.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

►

6.2.18 $$\mathrm{g}\left(z\right)=-\mathrm{Ci}\left(z\right)\cos z-\mathrm{si}\left(z\right)\sin z.$$

ⓘ

Defines:

$\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals

Symbols:

$\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{si}\left(z\right)$: sine integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.7

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

…

26: 13.13 Addition and Multiplication Theorems

… ►

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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

Annotations for §13.13(i), §13.13 and Ch.13

… ►

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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

Annotations for §13.13(ii), §13.13 and Ch.13

…

27: 13.2 Definitions and Basic Properties

… ►

13.2.2 $$M(a,b,z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s}{{\left(b\right)}_{s}s!}{z}^s=1+\frac{a}{b}z+\frac{a(a+1)}{b(b+1)2!}{z}^2+\mathrm{\cdots },$$

ⓘ

Defines:

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

Symbols:

${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, ${\left(a\right)}_n$: 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:

pMML, png, TeX

See also:

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

… ►

13.2.3 $$𝐌(a,b,z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s}{\mathrm{\Gamma }\left(b+s\right)s!}{z}^s,$$

ⓘ

Defines:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: 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:

pMML, png, TeX

See also:

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

… ►

13.2.5 $$\underset{b\to -n}{lim}\frac{M(a,b,z)}{\mathrm{\Gamma }\left(b\right)}=𝐌(a,-n,z)=\frac{{\left(a\right)}_{n+1}}{(n+1)!}{z}^{n+1}M(a+n+1,n+2,z).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E5

Encodings:

pMML, png, TeX

See also:

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

… ►

13.2.14 $$U(-n,b,z)={(-1)}^n{\left(b\right)}_n+O\left(z\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $U(a,b,z)$: 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/13.2.E14

Encodings:

pMML, png, TeX

See also:

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

►

13.2.15 $$U(-n+b-1,b,z)={(-1)}^n{\left(2-b\right)}_n{z}^{1-b}+O\left(z^{2-b}\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $U(a,b,z)$: 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/13.2.E15

Encodings:

pMML, png, TeX

See also:

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

…

28: 18.1 Notation

… ►

18.1.1 $$C_n^{(0)}\left(x\right)=\frac{2}{n}{T}_n\left(x\right)=\frac{2(n-1)!}{{\left(\frac{1}{2}\right)}_n}{P}_n^{(-\frac{1}{2},-\frac{1}{2})}\left(x\right),$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.1.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.1(iii), §18.1 and Ch.18

… ►

18.1.2 $$G_n(p,q,x)=\frac{n!}{{\left(n+p\right)}_n}{P}_n^{(p-q,q-1)}\left(2x-1\right),$$

ⓘ

Defines:

$G_n(p,q,x)$: shifted Jacobi polynomial

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $q$: real variable, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.1.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.1(iii), §18.1 and Ch.18

…

29: 16.10 Expansions in Series of ${}_p{}^{}F_q^{}$ Functions

… ►

16.10.1 $${}_{p+r}{}^{}F_{q+s}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,c_1,\mathrm{\dots },c_r}{b_1,\mathrm{\dots },b_q,d_1,\mathrm{\dots },d_s};z\zeta )=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(𝐚\right)}_k{\left(\alpha \right)}_k{\left(\beta \right)}_k{(-z)}^k}{{\left(𝐛\right)}_k{\left(\gamma +k\right)}_{k}k!}{}_{p+2}{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k,a_1+k,\mathrm{\dots },a_p+k}{\gamma +2k+1,b_1+k,\mathrm{\dots },b_q+k};z){}_{r+2}{}^{}F_{s+2}^{}(\genfrac{}{}{0pt}{}{-k,\gamma +k,c_1,\mathrm{\dots },c_r}{\alpha ,\beta ,d_1,\mathrm{\dots },d_s};\zeta ).$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.10 and Ch.16

… ►

16.10.2 $${}_{p+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{p+1}}{b_1,\mathrm{\dots },b_p};z\zeta )={(1-z)}^{-a_1}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a_1\right)}_k}{k!}{}_{p+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{-k,a_2,\mathrm{\dots },a_{p+1}}{b_1,\mathrm{\dots },b_p};\zeta ){\left(\frac{z}{z-1}\right)}^k.$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.10 and Ch.16

…

30: 13.6 Relations to Other Functions

… ►

13.6.11_1 $$M(\nu +\frac{1}{2},2\nu +1+n,2z)=\mathrm{\Gamma }\left(\nu \right){\mathrm{e}}^z{\left(z/2\right)}^{-\nu }\sum _{k=0}^n\frac{{\left(-n\right)}_k{\left(2\nu \right)}_k(\nu +k)}{{\left(2\nu +1+n\right)}_{k}k!}{I}_{\nu +k}\left(z\right),$$

ⓘ

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, $n$: nonnegative integer and $z$: complex variable

Source:

Prudnikov et al. (1990, page 579, (7.11.1.6))

Referenced by:

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

Permalink:

http://dlmf.nist.gov/13.6.E11_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

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

►

13.6.11_2 $$M(\nu +\frac{1}{2},2\nu +1-n,2z)=\mathrm{\Gamma }\left(\nu -n\right){\mathrm{e}}^z{\left(z/2\right)}^{n-\nu }\sum _{k=0}^n{(-1)}^k\frac{{\left(-n\right)}_k{\left(2\nu -2n\right)}_k(\nu -n+k)}{{\left(2\nu +1-n\right)}_{k}k!}{I}_{\nu +k-n}\left(z\right).$$

ⓘ

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, $n$: nonnegative integer and $z$: complex variable

Source:

Prudnikov et al. (1990, page 579, (7.11.1.7))

Referenced by:

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

Permalink:

http://dlmf.nist.gov/13.6.E11_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

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

… ►

13.6.19 $$U(-n,\alpha +1,z)={(-1)}^n{\left(\alpha +1\right)}_{n}M(-n,\alpha +1,z)={(-1)}^{n}n!L_n^{(\alpha )}\left(z\right).$$

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.6.9 and 13.6.27

Referenced by:

§13.6(v), §18.17(iv), §18.9(iii)

Permalink:

http://dlmf.nist.gov/13.6.E19

Encodings:

pMML, png, TeX

See also:

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

… ►

13.6.20 $$U(-n,z-n+1,a)={\left(-z\right)}_{n}M(-n,z-n+1,a)=a^n{C}_n(z;a).$$

ⓘ

Symbols:

$C_n(x;a)$: Charlier polynomial, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.6(v)

Permalink:

http://dlmf.nist.gov/13.6.E20

Encodings:

pMML, png, TeX

See also:

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

…