Leibniz formula for derivatives

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21: 14.21 Definitions and Basic Properties

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14.21.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }^2}{1-z^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.1

Referenced by:

§14.21(ii), §14.29, 2nd item

Permalink:

http://dlmf.nist.gov/14.21.E1

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See also:

Annotations for §14.21(i), §14.21 and Ch.14

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§14.21(iii) Properties

… ►This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …

22: 13.15 Recurrence Relations and Derivatives

§13.15 Recurrence Relations and Derivatives

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§13.15(ii) Differentiation Formulas

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13.15.15 $$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{\frac{1}{2}z}{z}^{\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(z\right)\right)={(-1)}^n{\left(-2\mu \right)}_n{\mathrm{e}}^{\frac{1}{2}z}{z}^{\mu -\frac{1}{2}(n+1)}{M}_{\kappa -\frac{1}{2}n,\mu -\frac{1}{2}n}\left(z\right),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E15

Encodings:

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See also:

Annotations for §13.15(ii), §13.15 and Ch.13

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13.15.16 $$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{\frac{1}{2}z}{z}^{-\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(z\right)\right)=\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n}{{\left(1+2\mu \right)}_n}{\mathrm{e}}^{\frac{1}{2}z}{z}^{-\mu -\frac{1}{2}(n+1)}{M}_{\kappa -\frac{1}{2}n,\mu +\frac{1}{2}n}\left(z\right),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E16

Encodings:

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See also:

Annotations for §13.15(ii), §13.15 and Ch.13

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13.15.26 $${\left(z\frac{d}{dz}z\right)}^n\left({\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa -1}{W}_{\kappa ,\mu }\left(z\right)\right)={(-1)}^n{\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa +n-1}{W}_{\kappa +n,\mu }\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E26

Encodings:

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See also:

Annotations for §13.15(ii), §13.15 and Ch.13

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23: 12.2 Differential Equations

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12.2.1 $$\frac{d^{2}w}{{dz}^2}+\left(a{z}^2+bz+c\right)w=0,$$

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Defines:

$a$: parameter (locally), $b$: parameter (locally) and $c$: parameter (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

19.1.1

Permalink:

http://dlmf.nist.gov/12.2.E1

Encodings:

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See also:

Annotations for §12.2(i), §12.2 and Ch.12

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12.2.2 $$\frac{d^{2}w}{{dz}^2}-\left(\frac{1}{4}{z}^2+a\right)w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $a$: parameter

A&S Ref:

19.1.2

Referenced by:

§12.10(i), §12.14(i), §12.17, §12.2(i), §12.2(i), §12.4, §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.2.E2

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See also:

Annotations for §12.2(i), §12.2 and Ch.12

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12.2.3 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{1}{4}{z}^2-a\right)w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $a$: parameter

A&S Ref:

19.1.3

Referenced by:

§12.14(vii), §12.14(ix), §12.14(i), §12.14(ix), §12.14(v), §12.14(x), §12.17, §12.2(i), §12.2(i)

Permalink:

http://dlmf.nist.gov/12.2.E3

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Annotations for §12.2(i), §12.2 and Ch.12

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§12.2(iv) Reflection Formulas

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§12.2(v) Connection Formulas

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24: 1.8 Fourier Series

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Parseval’s Formula

… ►at every point at which $f(x)$ has both a left-hand derivative (that is, (1.4.4) applies when $h\to 0-$) and a right-hand derivative (that is, (1.4.4) applies when $h\to 0+$). … … ►

§1.8(iv) Poisson’s Summation Formula

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1.8.16 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-{(n+x)}^2\omega }=\sqrt{\frac{\pi }{\omega }}\left(1+2\sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2{\pi }^2/\omega }\cos \left(2n\pi x\right)\right),$$ $\mathrm{\Re }\omega >0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $n$: nonnegative integer

Referenced by:

§1.8(iv)

Permalink:

http://dlmf.nist.gov/1.8.E16

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See also:

Annotations for §1.8(iv), §1.8 and Ch.1

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25: 24.17 Mathematical Applications

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Euler–Maclaurin Summation Formula

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Boole Summation Formula

… ►Let ${𝒮}_n$ denote the class of functions that have $n-1$ continuous derivatives on $ℝ$ and are polynomials of degree at most $n$ in each interval $(k,k+1)$, $k\in \mathbb{Z}$. …

26: Bibliography K

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Connection formulae for the first Painlevé transcendent in the complex domain. Lett. Math. Phys. 27 (4), pp. 243–252.

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External Links:

ISSN 0377-9017, Document, MathReview (B. L. J. Braaksma), ZentralBlatt

Referenced by:

§32.11(i), §32.12(i).

Encodings:

BibTeX

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The Hahn polynomials, formulas and an application. Scripta Math. 26, pp. 33–46.

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External Links:

MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§18.20(i), §18.21(ii), §18.26(ii).

Encodings:

BibTeX

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Asymptotic expansions of certain $q$-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.

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External Links:

ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§17.2(i).

Encodings:

BibTeX

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On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.15(iv).

Encodings:

BibTeX

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The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

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External Links:

Document, MathReview (M. Mikolas), ZentralBlatt

Referenced by:

(18.14.8), §18.14(i), §18.17(ii), §18.18(ii).

Encodings:

BibTeX

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27: 3.8 Nonlinear Equations

… ►For other efficient derivative-free methods, see Le (1985). … ►

§3.8(iv) Zeros of Polynomials

… ►Explicit formulas for the zeros are available if $n\le 4$; see §§1.11(iii) and 4.43. No explicit general formulas exist when $n\ge 5$. … ►

3.8.13 $$\frac{dz}{d\alpha }=-\frac{\partial f}{\partial \alpha }/\frac{\partial f}{\partial z}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/3.8.E13

Encodings:

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See also:

Annotations for §3.8(vi), §3.8 and Ch.3

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28: 16.8 Differential Equations

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16.8.1 $$\frac{d^{n}w}{{dz}^n}+f_{n-1}(z)\frac{d^{n-1}w}{{dz}^{n-1}}+f_{n-2}(z)\frac{d^{n-2}w}{{dz}^{n-2}}+\mathrm{\cdots }+f_1(z)\frac{dw}{dz}+f_0(z)w=0$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $f_j(z)$: coefficients

Referenced by:

§16.8(i)

Permalink:

http://dlmf.nist.gov/16.8.E1

Encodings:

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See also:

Annotations for §16.8(i), §16.8 and Ch.16

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$𝐷={\displaystyle \frac{d}{dz}},$

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$\mathit{\vartheta }=z{\displaystyle \frac{d}{dz}},$

… ►We have the connection formula … ►Analytical continuation formulas for ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$ near $z=1$ are given in Bühring (1987b) for the case $q=2$, and in Bühring (1992) for the general case. …

29: 24.4 Basic Properties

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§24.4(v) Multiplication Formulas

… ►Next, … ►

§24.4(vii) Derivatives

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24.4.34 $$\frac{d}{dx}{B}_n\left(x\right)=n{B}_{n-1}\left(x\right),$$ $n=1,2,\mathrm{\dots }$,

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer and $x$: real or complex

A&S Ref:

23.1.5

Referenced by:

§24.13(i), (25.11.7)

Permalink:

http://dlmf.nist.gov/24.4.E34

Encodings:

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See also:

Annotations for §24.4(vii), §24.4 and Ch.24

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24.4.35 $$\frac{d}{dx}{E}_n\left(x\right)=n{E}_{n-1}\left(x\right),$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer and $x$: real or complex

A&S Ref:

23.1.5

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.4.E35

Encodings:

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See also:

Annotations for §24.4(vii), §24.4 and Ch.24

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30: 18.5 Explicit Representations

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§18.5(ii) Rodrigues Formulas

… ►Related formula: …See (Erdélyi et al., 1953b, §10.9(37)) for a related formula for ultraspherical polynomials. … ►and two similar formulas by symmetry; compare the second row in Table 18.6.1. … ►For corresponding formulas for Chebyshev, Legendre, and the Hermite ${\mathrm{𝐻𝑒}}_n$ polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …