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21: 13.3 Recurrence Relations and Derivatives

… ►

13.3.16 $$\frac{d^n}{{dz}^n}M(a,b,z)=\frac{{\left(a\right)}_n}{{\left(b\right)}_n}M(a+n,b+n,z),$$

ⓘ

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), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.4.9

Referenced by:

§13.3(i)

Permalink:

http://dlmf.nist.gov/13.3.E16

Encodings:

pMML, png, TeX

See also:

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

►

13.3.17 $${\left(z\frac{d}{dz}z\right)}^n\left(z^{a-1}M(a,b,z)\right)={\left(a\right)}_n{z}^{a+n-1}M(a+n,b,z),$$

ⓘ

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), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.3.E17

Encodings:

pMML, png, TeX

See also:

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

… ►

13.3.23 $$\frac{d^n}{{dz}^n}U(a,b,z)={(-1)}^n{\left(a\right)}_{n}U(a+n,b+n,z),$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.4.22

Referenced by:

§13.3(i)

Permalink:

http://dlmf.nist.gov/13.3.E23

Encodings:

pMML, png, TeX

See also:

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

… ►

13.3.25 $$\frac{d^n}{{dz}^n}\left(z^{b-1}U(a,b,z)\right)={(-1)}^n{\left(a-b+1\right)}_n{z}^{b-n-1}U(a,b-n,z),$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.3.E25

Encodings:

pMML, png, TeX

See also:

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

… ►Other versions of several of the identities in this subsection can be constructed with the aid of the operator identity …

22: 1.6 Vectors and Vector-Valued Functions

… ►

Levi-Civita Symbol

►

1.6.14

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Defines:

${ϵ}_{jk\mathrm{\ell }}$: Levi-Civita symbol

Symbols:

$j$: integer and $k$: integer

Permalink:

http://dlmf.nist.gov/1.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

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1.6.17 $${𝐞}_j\times {𝐞}_k={ϵ}_{jk\mathrm{\ell }}{𝐞}_{\mathrm{\ell }};$$

ⓘ

Symbols:

${ϵ}_{jk\mathrm{\ell }}$: Levi-Civita symbol, $j$: integer, $k$: integer and ${𝐞}_j$: unit vectors

Permalink:

http://dlmf.nist.gov/1.6.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

… ►

Del Operator

►

1.6.19 $$\nabla =𝐢\frac{\partial }{\partial x}+𝐣\frac{\partial }{\partial y}+𝐤\frac{\partial }{\partial z}.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: variable, $𝐢$: unit vector, $𝐣$: unit vector and $𝐤$: unit vector

Permalink:

http://dlmf.nist.gov/1.6.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

…

23: 16.3 Derivatives and Contiguous Functions

… ►

16.3.1 $$\frac{d^n}{{dz}^n}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)=\frac{{\left(𝐚\right)}_n}{{\left(𝐛\right)}_n}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1+n,\mathrm{\dots },a_p+n}{b_1+n,\mathrm{\dots },b_q+n};z),$$

ⓘ

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), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $p$: 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/16.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.3(i), §16.3 and Ch.16

►

16.3.2 $$\frac{d^n}{{dz}^n}\left(z^{\gamma }{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)\right)={\left(\gamma -n+1\right)}_n{z}^{\gamma -n}{}_{p+1}{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{\gamma +1,a_1,\mathrm{\dots },a_p}{\gamma +1-n,b_1,\mathrm{\dots },b_q};z),$$

ⓘ

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), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $p$: 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/16.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.3(i), §16.3 and Ch.16

►

16.3.3 $${\left(z\frac{d}{dz}z\right)}^n\left(z^{\gamma -1}{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{\gamma ,a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)\right)={\left(\gamma \right)}_n{z}^{\gamma +n-1}{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{\gamma +n,a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z),$$

ⓘ

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), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $p$: 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/16.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.3(i), §16.3 and Ch.16

►

16.3.4 $$\frac{d^n}{{dz}^n}\left(z^{\gamma -1}{}_p{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{\gamma ,b_1,\mathrm{\dots },b_q};z)\right)={\left(\gamma -n\right)}_n{z}^{\gamma -n-1}{}_p{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{\gamma -n,b_1,\mathrm{\dots },b_q};z).$$

ⓘ

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), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $p$: 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/16.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.3(i), §16.3 and Ch.16

►Other versions of these identities can be constructed with the aid of the operator identity …

24: 18.19 Hahn Class: Definitions

… ►The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator $\frac{d}{dx}$ in the case of the classical OP’s is played by a suitable difference operator. … ►

Hahn class (or linear lattice class). These are OP’s $p_n(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta }}_x$ or ${\nabla }_x$ or ${\delta }_x$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

►

Wilson class (or quadratic lattice class). These are OP’s $p_n(x)=p_n(\lambda (y))$ ($p_n(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

… ►

18.19.5 $$k_n=\frac{{\left(n+2\mathrm{\Re }\left(a+b\right)-1\right)}_n}{n!}.$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $k_n$

Referenced by:

§18.19

Permalink:

http://dlmf.nist.gov/18.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

…

25: 18.26 Wilson Class: Continued

… ►

18.26.2 $$S_n(y^2;a,b,c)={\left(a+b\right)}_n{\left(a+c\right)}_n{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,a+iy,a-iy}{a+b,a+c};1).$$

ⓘ

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), $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $y$: real variable and $n$: nonnegative integer

Referenced by:

§18.26(i), §18.26(ii), §18.26(iv)

Permalink:

http://dlmf.nist.gov/18.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(i), §18.26 and Ch.18

… ►

18.26.5 $$\underset{d\to \mathrm{\infty }}{lim}\frac{W_n(x;a,b,c,d)}{{\left(a+d\right)}_n}=S_n(x;a,b,c).$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_n(x;a,b,c,d)$: Wilson polynomial, $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

… ►For comments on the use of the forward-difference operator ${\mathrm{\Delta }}_x$, the backward-difference operator ${\nabla }_x$, and the central-difference operator ${\delta }_x$, see §18.2(ii). … ►

18.26.16 $$\frac{{\mathrm{\Delta }}_y\left(R_n(y(y+\gamma +\delta +1);\alpha ,\beta ,\gamma ,\delta )\right)}{{\mathrm{\Delta }}_y\left(y(y+\gamma +\delta +1)\right)}=\frac{n(n+\alpha +\beta +1)}{(\alpha +1)(\beta +\delta +1)(\gamma +1)}{R}_{n-1}(y(y+\gamma +\delta +2);\alpha +1,\beta +1,\gamma +1,\delta ).$$

ⓘ

Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $\mathrm{\Delta }$: forward difference operator, $y$: real variable, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§16.4(iii)

Permalink:

http://dlmf.nist.gov/18.26.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(iii), §18.26 and Ch.18

►

18.26.17 $$\frac{{\mathrm{\Delta }}_y\left(R_n(y(y+\gamma +\delta +1);\gamma ,\delta ,N)\right)}{{\mathrm{\Delta }}_y\left(y(y+\gamma +\delta +1)\right)}=-\frac{n}{(\gamma +1)N}{R}_{n-1}(y(y+\gamma +\delta +2);\gamma +1,\delta ,N-1).$$

ⓘ

Symbols:

$R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $\mathrm{\Delta }$: forward difference operator, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§18.26(iii)

Permalink:

http://dlmf.nist.gov/18.26.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(iii), §18.26 and Ch.18

…

26: Bibliography K

… ►

Lowering and Raising Operators for Some Special Orthogonal Polynomials. In Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, pp. 227–238.

ⓘ

External Links:

FullText, MathReview Entry, ZentralBlatt

Referenced by:

§18.9(iii).

Encodings:

BibTeX

… ►

Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

Encodings:

BibTeX

►

Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

ⓘ

External Links:

ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

Encodings:

BibTeX

… ►

Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces. Funktsional. Anal. i Prilozhen. 23 (1), pp. 24–40 (Russian).

ⓘ

Notes:

In Russian. English translation: Funct. Anal. Appl. 23(1989), no. 1, pp. 19–33

External Links:

ISSN 0374-1990, Document, MathReview (Fedor Malikov), ZentralBlatt

Referenced by:

§20.12(ii).

Encodings:

BibTeX

… ►

Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions. J. Math. Anal. Appl. 324 (1), pp. 285–303.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§18.36(i).

Encodings:

BibTeX

27: 18.28 Askey–Wilson Class

… ► $y$) such that $P_n(z)=p_n(\frac{1}{2}(z+z^{-1}))$ in the Askey–Wilson case, and $P_n(y)=p_n(q^{-y}+c{q}^{y+1})$ in the $q$-Racah case, and both are eigenfunctions of a second order $q$-difference operator similar to (18.27.1). … ►

18.28.4 $$h_0=\frac{{(abcd;q)}_{\mathrm{\infty }}}{{(q,ab,ac,ad,bc,bd,cd;q)}_{\mathrm{\infty }}},$$

ⓘ

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 and $h_n$

Permalink:

http://dlmf.nist.gov/18.28.E4

Encodings:

pMML, png, TeX

See also:

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

… ►where the operator $L$ is defined by … ►In Tsujimoto et al. (2012) an extension of the Bannai–Ito polynomials occurs as eigenfunctions of a Dunkl type operator. …

28: 31.10 Integral Equations and Representations

… ►

31.10.3 $$({𝒟}_z-{𝒟}_t)𝒦=0,$$

ⓘ

Defines:

$𝒦(z,t)$: kernel (locally)

Symbols:

$z$: complex variable and ${𝒟}_z$: Heun’s operator

Permalink:

http://dlmf.nist.gov/31.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10 and Ch.31

►where ${𝒟}_z$ is Heun’s operator in the variable $z$ : … ►The solutions of (31.10.8) are given in terms of the Riemann $P$-symbol (see §15.11(i)) as … ►For suitable choices of the branches of the $P$-symbols in (31.10.9) and the contour $C$, we can obtain both integral equations satisfied by Heun functions, as well as the integral representations of a distinct solution of Heun’s equation in terms of a Heun function (polynomial, path-multiplicative solution). … ►

31.10.14 $$\left((t-z){𝒟}_s+(z-s){𝒟}_t+(s-t){𝒟}_z\right)𝒦=0,$$

ⓘ

Defines:

$𝒦(z;s,t)$: kernel (locally)

Symbols:

$z$: complex variable and ${𝒟}_z$: Heun’s operator

Permalink:

http://dlmf.nist.gov/31.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(ii), §31.10 and Ch.31

…

29: Bibliography G

… ►

Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, Vol. 4, Kluwer Academic Publishers, Dordrecht.

ⓘ

External Links:

ISBN 1-4020-0101-0, MathReview Entry

Referenced by:

Profile Mourad E.H. Ismail.

Encodings:

BibTeX

… ►

Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.

ⓘ

Notes:

Translated from the Russian by Eugene Saletan. A paperbound reprinting by the same publisher appeared in 1977

External Links:

MathReview, ZentralBlatt

Referenced by:

§2.6(i).

Encodings:

BibTeX

… ►

WKB and turning point theory for second-order difference equations. In Spectral Methods for Operators of Mathematical Physics, Oper. Theory Adv. Appl., Vol. 154, pp. 101–138.

ⓘ

External Links:

FullText, MathReview (Ioannis K. Purnaras), ZentralBlatt

Referenced by:

§18.15(v), §18.15(v).

Encodings:

BibTeX

… ►

Riemann surfaces, plane algebraic curves and their period matrices. J. Symbolic Comput. 26 (6), pp. 789–803.

ⓘ

External Links:

ISSN 0747-7171, Document, MathReview (Gabino González Díez), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

A new identity for some six-$j$ symbols. J. Math. Phys. 32 (6), pp. 1430–1432.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview

Referenced by:

§34.5(vi).

Encodings:

BibTeX

…

30: Mathematical Introduction

… ► ►►►►

$ℂ$	complex plane (excluding infinity).
…
$\mathrm{\Delta }$ (or ${\mathrm{\Delta }}_x$)	forward difference operator: $\mathrm{\Delta }f(x)=f(x+1)-f(x)$.
$\nabla$ (or ${\nabla }_x$)	backward difference operator: $\nabla f(x)=f(x)-f(x-1)$. (See also del operator in the Notations section.)
…

► ►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
${\left(\alpha \right)}_n$	Pochhammer’s symbol: $\alpha (\alpha +1)(\alpha +2)\mathrm{\cdots }(\alpha +n-1)$ if $n=1,2,3,\mathrm{\dots }$; 1 if $n=0$.
…

…