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31—40 of 209 matching pages

31: 8.15 Sums

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8.15.2 $$a\sum _{k=1}^{\mathrm{\infty }}\left(\frac{{\mathrm{e}}^{2\pi \mathrm{i}k(z+h)}}{{\left(2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,2\pi \mathrm{i}kz)+\frac{{\mathrm{e}}^{-2\pi \mathrm{i}k(z+h)}}{{\left(-2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,-2\pi \mathrm{i}kz)\right)=\zeta (-a,z+h)+\frac{z^{a+1}}{a+1}+\left(h-\frac{1}{2}\right)z^a,$$ $h\in [0,1]$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Proof sketch:

Start with and $z>0$. For $\mathrm{\Gamma }(a,\pm 2\pi \mathrm{i}kz)$ take integral representation (8.2.2) and use the substitution $t=\pm 2\pi \mathrm{i}kz(1+\tau )$. The sum and the integral can be interchanged, and the sum can be evaluated via (4.6.1). Use integration by parts. This will result in $\left(h-\frac{1}{2}\right)z^a$ plus two integrals with infinitely many poles. The residue theorem (§1.10(iv)) will give us an infinite series which can be identified via (25.11.1). For other values of $a$ and $z$ use analytic continuation.

Referenced by:

§25.11(x), §25.11(x), §8.15, Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/8.15.E2

Encodings:

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Addition (effective with 1.1.3):

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See also:

Annotations for §8.15 and Ch.8

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32: 8.28 Software

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33: 10.77 Software

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34: 16.6 Transformations of Variable

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35: 18.35 Pollaczek Polynomials

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18.35.2_1 $$x{P}_n^{(\lambda )}(x;a,b,c)=\frac{n+c+1}{2(n+c+\lambda +a)}{P}_{n+1}^{(\lambda )}(x;a,b,c)-\frac{b}{n+c+\lambda +a}{P}_n^{(\lambda )}(x;a,b,c)+\frac{n+c+2\lambda -1}{2(n+c+\lambda +a)}{P}_{n-1}^{(\lambda )}(x;a,b,c),$$ $n=0,1,\mathrm{\dots }$.

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Symbols:

$Q_n^{(\lambda )}(x;a,b,c)$: Pollaczek polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.35(i), Erratum (V1.2.0) Section 18.35

Permalink:

http://dlmf.nist.gov/18.35.E2_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

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See also:

Annotations for §18.35(i), §18.35 and Ch.18

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18.35.2_2 $$Q_n^{(\lambda )}(x;a,b,c)=\frac{{\left(c+1\right)}_n}{2^n{\left(c+\lambda +a\right)}_n}{P}_n^{(\lambda )}(x;a,b,c)$$

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Defines:

$Q_n^{(\lambda )}(x;a,b,c)$: Pollaczek polynomial

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.35(ii)

Permalink:

http://dlmf.nist.gov/18.35.E2_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.35(i), §18.35 and Ch.18

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18.35.2_5 $$P_n^{(\lambda )}(-x;a,b,c)={(-1)}^n{P}_n^{(\lambda )}(x;a,-b,c).$$

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Symbols:

$Q_n^{(\lambda )}(x;a,b,c)$: Pollaczek polynomial, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Use (18.35.1), (18.35.2).

Referenced by:

§18.35(i), §18.35(ii), Erratum (V1.2.0) Section 18.35

Permalink:

http://dlmf.nist.gov/18.35.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.35(i), §18.35 and Ch.18

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18.35.6_2

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Symbols:

$n$: nonnegative integer and $x$: real variable

Proved:

Ismail (2009, (5.5.8)); Case (iii) is overlooked in the given reference.(proved)

Proof sketch:

By Favard’s theorem (see §18.2(viii)) the necessary and sufficient condition for orthogonality is that in (18.35.2_4), for $c=0$, the coefficient of $Q_n^{(\lambda )}(x;a,b,0)$ is real for $n\ge 0$ and the coefficient of $Q_{n-1}^{(\lambda )}(x;a,b,0)$ is positive for $n\ge 1$.

Referenced by:

§18.35(ii), Erratum (V1.2.0) for Equation (18.35.5)

Permalink:

http://dlmf.nist.gov/18.35.E6_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

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18.35.10 $${𝒫}_n^{\lambda }(x;\varphi ,c)=P_n^{(\lambda )}(\cos \varphi ;0,x\sin \varphi ,c).$$

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Symbols:

$Q_n^{(\lambda )}(x;a,b,c)$: Pollaczek polynomial, ${𝒫}_n^{\lambda }(x;\varphi ,c)$: associated Meixner–Pollaczek polynomial, $\cos z$: cosine function, $\sin z$: sine function, $n$: nonnegative integer and $x$: real variable

Source:

Askey and Wimp (1984)

Referenced by:

§18.30(v), §18.35(iii), Erratum (V1.2.0) Section 18.35

Permalink:

http://dlmf.nist.gov/18.35.E10

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.35(iii), §18.35 and Ch.18

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36: 18.2 General Orthogonal Polynomials

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18.2.9_5 $$A_{n-1}{A}_n{C}_n>0,$$ $n\ge 1$.

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Symbols:

$n$: nonnegative integer, $C_n$: real constant and $A_n$: real constant

Referenced by:

§18.2(iv), §18.2(iv), §18.2(viii), §18.30(vi), §18.36(v), Erratum (V1.2.0) Section 18.2

Permalink:

http://dlmf.nist.gov/18.2.E9_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

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18.2.11_2 $$a_{n-1}{c}_n>0,$$ $n\ge 1$.

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Symbols:

$n$: nonnegative integer, $a_n$: real constant and $c_n$: real constant

Referenced by:

§18.2(viii)

Permalink:

http://dlmf.nist.gov/18.2.E11_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

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See also:

Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

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18.2.11_6 $${\beta }_n>0,$$ $n\ge 1$.

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Symbols:

$n$: nonnegative integer

Referenced by:

§18.2(viii)

Permalink:

http://dlmf.nist.gov/18.2.E11_6

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

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18.2.11_7 $$q_n(x)=p_n(x)/\sqrt{h_n},$$ $n\ge 0$.

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Symbols:

$p_n(x)$: polynomial of degree $n$ , $q$: real variable, $n$: nonnegative integer, $x$: real variable and $h_n$

Permalink:

http://dlmf.nist.gov/18.2.E11_7

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

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See also:

Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

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18.2.16 $$q_n(x)-q_{n-1}(x)=\frac{p_n(y)}{h_n}{p}_n(x),$$

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Symbols:

$y$: real variable, $p_n(x)$: polynomial of degree $n$ , $q$: real variable, $n$: nonnegative integer, $x$: real variable and $h_n$

Referenced by:

§18.9(ii)

Permalink:

http://dlmf.nist.gov/18.2.E16

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.2(v), §18.2(v), §18.2 and Ch.18

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37: 18.27 $q$-Hahn Class

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18.27.4_1 $$h_n=\frac{{\left(\alpha q\right)}^{nN}}{1-\alpha \beta q^{2n+1}}\frac{{(\alpha \beta q^{n+1};q)}_{N+1}{(\beta q;q)}_n}{{\left[\genfrac{}{}{0.0pt}{}{N}{n}\right]}_q{(\alpha q;q)}_n}.$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $h_n$

Referenced by:

§18.27(ii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(ii), §18.27 and Ch.18

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18.27.4_2 $$\underset{q\to 1}{lim}{Q}_n(q^{-x};q^{\alpha },q^{\beta },N;q)=Q_n(x;\alpha ,\beta ,N).$$

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $Q_n(x;\alpha ,\beta ,N;q)$: $q$-Hahn polynomial, $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.27(ii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E4_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(ii), §18.27 and Ch.18

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18.27.6_5 $$P_n(x;a,b,c,d;q)=P_n(qa{c}^{-1}x;a,b,-a{c}^{-1}d;q).$$

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Symbols:

$P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer, $x$: real variable and $P_n^{(\alpha ,\beta )}(x;c,d;q)$

Source:

Koekoek et al. (2010, p. 442)

Referenced by:

§18.27(iii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E6_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(iii), §18.27 and Ch.18

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18.27.14_1 $$h_n=\frac{{\left(aq\right)}^n}{1-ab{q}^{2n+1}}\frac{{(q,bq;q)}_n}{{(aq;q)}_n}\frac{{(ab{q}^{n+1};q)}_{\mathrm{\infty }}}{{(aq;q)}_{\mathrm{\infty }}}.$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $h_n$

Source:

Koekoek et al. (2010, (14.12.2))

Referenced by:

§18.27(iv), (18.28.27), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E14_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(iv), §18.27 and Ch.18

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18.27.14_6 $$\underset{q\to 1}{lim}{p}_n((1-q)x;q^{\alpha },0;q)=\frac{n!}{{\left(\alpha +1\right)}_n}{L}_n^{(\alpha )}\left(x\right).$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p_n(x;a,b;q)$: little $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.12.12), (14.20.12))

Referenced by:

§18.27(iv), §18.27(iv), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E14_6

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(iv), §18.27(iv), §18.27 and Ch.18

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38: 5.6 Inequalities

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39: 16.13 Appell Functions

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40: 18.28 Askey–Wilson Class

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18.28.6_1 $$(L{R}_n)(z)=(q^{-n}+abcd{q}^{n-1})R_n(z),$$

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Symbols:

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

Referenced by:

§18.28(ii), §18.28(ii), Erratum (V1.2.0) Section 18.28

Permalink:

http://dlmf.nist.gov/18.28.E6_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

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18.28.6_2 $$(Lf)(z)=\frac{(1-az)(1-bz)(1-cz)(1-dz)}{(1-z^2)(1-q{z}^2)}\left(f(qz)-f(z)\right)+\frac{(1-a{z}^{-1})(1-b{z}^{-1})(1-c{z}^{-1})(1-d{z}^{-1})}{(1-z^{-2})(1-q{z}^{-2})}\left(f(q^{-1}z)-f(z)\right)+\left(1+q^{-1}abcd\right)f(z).$$

ⓘ

Symbols:

$z$: complex variable and $q$: real variable

Referenced by:

§18.28(ii), §18.38(iii), §18.38(iii)

Permalink:

http://dlmf.nist.gov/18.28.E6_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

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18.28.6_3 $$(z+z^{-1})R_n(z)=a_n\left(R_{n+1}(z)-R_n(z)\right)+c_n\left(R_{n-1}(z)-R_n(z)\right)+(a+a^{-1})R_n(z),$$

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Symbols:

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

Referenced by:

§18.28(ii)

Permalink:

http://dlmf.nist.gov/18.28.E6_3

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

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18.28.6_5 $$R_n(a^{-1}{q}^{-m};a,b,c,d|q)=R_m({\tilde{a}}^{-1}{q}^{-n};\tilde{a},\tilde{b},\tilde{c},\tilde{d}|q),$$ $m,n=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$q$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.28(ii), §18.28(ii), Erratum (V1.2.0) Section 18.28

Permalink:

http://dlmf.nist.gov/18.28.E6_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

… ►More generally, if $|ab|\le 1$ instead of $|a|,|b|\le 1$, discrete terms need to be added to the right-hand side of (18.28.8); see Koekoek et al. (2010, Eq. (14.8.3)). …