Jacobi symbol

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1: 27.9 Quadratic Characters

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27.9.3 $$(p|q)(q|p)={(-1)}^{(p-1)(q-1)/4}.$$

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Symbols:

$(n|p)$: Legendre symbol, $p$: odd prime and $q$: odd prime

Referenced by:

§27.9

Permalink:

http://dlmf.nist.gov/27.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.9 and Ch.27

►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. The Jacobi symbol $(n|P)$ is a Dirichlet character (mod $P$). …

2: 27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
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$(n|P)$	Jacobi symbol; see §27.9.
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3: 20.1 Special Notation

… ►Primes on the $\theta$ symbols indicate derivatives with respect to the argument of the $\theta$ function. …

4: 18.30 Associated OP’s

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18.30.5 $$\frac{{(-1)}^n{\left(\alpha +\beta +c+1\right)}_{n}n!P_n^{(\alpha ,\beta )}(x;c)}{{\left(\alpha +\beta +2c+1\right)}_n{\left(\beta +c+1\right)}_n}=\sum _{\mathrm{\ell }=0}^n\frac{{\left(-n\right)}_{\mathrm{\ell }}{\left(n+\alpha +\beta +2c+1\right)}_{\mathrm{\ell }}}{{\left(c+1\right)}_{\mathrm{\ell }}{\left(\beta +c+1\right)}_{\mathrm{\ell }}}{\left(\frac{1}{2}x+\frac{1}{2}\right)}^{\mathrm{\ell }}{}_4{}^{}F_3^{}(\genfrac{}{}{0pt}{}{\mathrm{\ell }-n,n+\mathrm{\ell }+\alpha +\beta +2c+1,\beta +c,c}{\beta +\mathrm{\ell }+c+1,\mathrm{\ell }+c+1,\alpha +\beta +2c};1),$$

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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), $P_n^{(\alpha ,\beta )}(x;c)$: associated Jacobi polynomial, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.30

Permalink:

http://dlmf.nist.gov/18.30.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(i), §18.30 and Ch.18

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5: 18.12 Generating Functions

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18.12.2_5 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\gamma ,\alpha +\beta +1-\gamma }{\alpha +1};\frac{1-R-z}{2}){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\gamma ,\alpha +\beta +1-\gamma }{\beta +1};\frac{1-R+z}{2})=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\gamma \right)}_n{\left(\alpha +\beta +1-\gamma \right)}_n}{{\left(\alpha +1\right)}_n{\left(\beta +1\right)}_n}{P}_n^{(\alpha ,\beta )}\left(x\right)z^n,$$ $R=\sqrt{1-2xz+z^2}$, ,

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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), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proved:

Rainville (1960, §141, (2)); Brafman’s generating function(proved)

Source:

Koekoek et al. (2010, (9.8.15))

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) §18.12

Permalink:

http://dlmf.nist.gov/18.12.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.3 $${(1+z)}^{-\alpha -\beta -1}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}(\alpha +\beta +1),\frac{1}{2}(\alpha +\beta +2)}{\beta +1};\frac{2(x+1)z}{{(1+z)}^2})=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\alpha +\beta +1\right)}_n}{{\left(\beta +1\right)}_n}{P}_n^{(\alpha ,\beta )}\left(x\right)z^n,$$ ,

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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), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.12.3_5), §18.12, §18.12

Permalink:

http://dlmf.nist.gov/18.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.3_5 $$\frac{1+z}{{(1-2xz+z^2)}^{\beta +\frac{3}{2}}}=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(2\beta +2\right)}_n}{{\left(\beta +1\right)}_n}{P}_n^{(\beta +1,\beta )}\left(x\right)z^n,$$ ,

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Symbols:

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

Proof sketch:

Substitute $\alpha =\beta +1$ in (18.12.3) and use (15.4.6).

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) §18.12

Permalink:

http://dlmf.nist.gov/18.12.E3_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.4 $${(1-2xz+z^2)}^{-\lambda }=\sum _{n=0}^{\mathrm{\infty }}{C}_n^{(\lambda )}\left(x\right)z^n=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(2\lambda \right)}_n}{{\left(\lambda +\frac{1}{2}\right)}_n}{P}_n^{(\lambda -\frac{1}{2},\lambda -\frac{1}{2})}\left(x\right)z^n,$$ .

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

A&S Ref:

22.9.3

Referenced by:

§18.12, §18.18(viii)

Permalink:

http://dlmf.nist.gov/18.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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6: 18.7 Interrelations and Limit Relations

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

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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

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

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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

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18.7.15 $$C_{2n}^{(\lambda )}\left(x\right)=\frac{{\left(\lambda \right)}_n}{{\left(\frac{1}{2}\right)}_n}{P}_n^{(\lambda -\frac{1}{2},-\frac{1}{2})}\left(2x^2-1\right),$$

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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:

22.5.25

Referenced by:

§18.7(ii), §18.9(ii)

Permalink:

http://dlmf.nist.gov/18.7.E15

Encodings:

pMML, png, TeX

See also:

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

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18.7.16 $$C_{2n+1}^{(\lambda )}\left(x\right)=\frac{{\left(\lambda \right)}_{n+1}}{{\left(\frac{1}{2}\right)}_{n+1}}x{P}_n^{(\lambda -\frac{1}{2},\frac{1}{2})}\left(2x^2-1\right).$$

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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:

22.5.26

Referenced by:

§18.7(ii), §18.9(ii)

Permalink:

http://dlmf.nist.gov/18.7.E16

Encodings:

pMML, png, TeX

See also:

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

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7: 18.1 Notation

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

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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

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

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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

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8: 18.14 Inequalities

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18.14.1 $$|P_n^{(\alpha ,\beta )}\left(x\right)|\le P_n^{(\alpha ,\beta )}\left(1\right)=\frac{{\left(\alpha +1\right)}_n}{n!},$$ $-1\le x\le 1$, $\alpha \ge \beta >-1$, $\alpha \ge -\frac{1}{2}$,

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Symbols:

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

A&S Ref:

22.14.1

Referenced by:

§18.14(i)

Permalink:

http://dlmf.nist.gov/18.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

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18.14.2 $$|P_n^{(\alpha ,\beta )}\left(x\right)|\le |P_n^{(\alpha ,\beta )}\left(-1\right)|=\frac{{\left(\beta +1\right)}_n}{n!},$$ $-1\le x\le 1$, $\beta \ge \alpha >-1$, $\beta \ge -\frac{1}{2}$.

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Symbols:

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

A&S Ref:

22.14.1

Referenced by:

§18.14(i)

Permalink:

http://dlmf.nist.gov/18.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

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18.14.3_5 $${\left(\frac{1}{2}(1+x)\right)}^{\beta /2}\left|P_n^{(\alpha ,\beta )}\left(x\right)\right|\le P_n^{(\alpha ,\beta )}\left(1\right)=\frac{{\left(\alpha +1\right)}_n}{n!},$$ $-1\le x\le 1$, $\alpha ,\beta \ge 0$.

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Symbols:

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

Proved:

Carnicer et al. (2020, Theorem 2)(proved)

Proof sketch:

The case $\alpha =0$ follows from the result on monotonic weight functions in §18.2(xii).

Referenced by:

§18.14(i), §18.14(i), §18.2(xii), Erratum (V1.2.0) §18.14

Permalink:

http://dlmf.nist.gov/18.14.E3_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

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18.14.25 $$\sum _{m=0}^n\frac{{\left(\lambda +1\right)}_{n-m}}{(n-m)!}\frac{{\left(\lambda +1\right)}_m}{m!}\frac{P_m^{(\alpha ,\beta )}\left(x\right)}{P_m^{(\beta ,\alpha )}\left(1\right)}\ge 0,$$ $x\ge -1$, $\alpha +\beta \ge \lambda \ge 0$, $\beta \ge -\frac{1}{2}$, $n=0,1,\mathrm{\dots }$.

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Symbols:

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

Proved:

Gasper (1977, Theorem 5)(proved)

Source:

Askey and Gasper (1976, Conjecture 1); conjectured

Referenced by:

Erratum (V1.2.0) §18.14

Permalink:

http://dlmf.nist.gov/18.14.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18

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9: 18.11 Relations to Other Functions

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18.11.1 $${𝖯}_n^m\left(x\right)={\left(\frac{1}{2}\right)}_m{(-2)}^m{(1-x^2)}^{\frac{1}{2}m}{C}_{n-m}^{(m+\frac{1}{2})}\left(x\right)={\left(n+1\right)}_m{(-2)}^{-m}{(1-x^2)}^{\frac{1}{2}m}{P}_{n-m}^{(m,m)}\left(x\right),$$ $0\le m\le n$.

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Symbols:

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

Referenced by:

§14.3(iv), §18.11(i)

Permalink:

http://dlmf.nist.gov/18.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

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10: 18.5 Explicit Representations

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

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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

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

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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

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