integer degree and order

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31: Bibliography

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Polygamma functions of negative order. J. Comput. Appl. Math. 100 (2), pp. 191–199.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

(25.11.32), (25.11.34), §25.11(viii).

Encodings:

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On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

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Congruences of $p$-adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.

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External Links:

ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Algorithm 804: Subroutines for the computation of Mathieu functions of integer orders. ACM Trans. Math. Software 26 (3), pp. 408–414.

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Notes:

Double-precision C++, minimum accuracy 14D

External Links:

ISSN 0098-3500, Document, GAMS (module 804; package TOMS)

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Spherical Bessel functions $j_n$ and $y_n$ of integer order and real argument. Comput. Phys. Comm. 14 (3-4), pp. 261–265.

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Notes:

Single-precision Fortran, maximum accuracy 10S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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32: 24.16 Generalizations

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§24.16(i) Higher-Order Analogs

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Polynomials and Numbers of Integer Order

►For $\mathrm{\ell }=0,1,2,\mathrm{\dots }$, Bernoulli and Euler polynomials of order $\mathrm{\ell }$ are defined respectively by … ► $B_n^{(x)}$ is a polynomial in $x$ of degree $n$. … ►In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); $p$-adic integer order Bernoulli numbers (Adelberg (1996)); $p$-adic $q$-Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).

33: 18.39 Applications in the Physical Sciences

… ►The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18. … ►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ► ${ϵ}_0$ is referred to as the ground state, all others, $n=1,2,\mathrm{\dots },$ in order of increasing energy being excited states. … ►Namely the $k$th eigenfunction, listed in order of increasing eigenvalues, starting at $k=0$, has precisely $k$ nodes, as real zeros of wave-functions away from boundaries are often referred to. … ►Physical scientists use the $n$ of Bohr as, to $0$th and $1$st order, it describes the structure and organization of the Periodic Table of the Chemical Elements of which the Hydrogen atom is only the first. …

34: 23.20 Mathematical Applications

… ►Let $T$ denote the set of points on $C$ that are of finite order (that is, those points $P$ for which there exists a positive integer $n$ with $nP=o$), and let $I,K$ be the sets of points with integer and rational coordinates, respectively. …The resulting points are then tested for finite order as follows. …If any of these quantities is zero, then the point has finite order. If any of $2P$, $4P$, $8P$ is not an integer, then the point has infinite order. … ►The modular equation of degree $p$, $p$ prime, is an algebraic equation in $\alpha =\lambda \left(p\tau \right)$ and $\beta =\lambda \left(\tau \right)$. …

35: 13.2 Definitions and Basic Properties

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Kummer’s Equation

… ►When $a=-n$, $n=0,1,2,\mathrm{\dots }$, $𝐌(a,b,z)$ is a polynomial in $z$ of degree not exceeding $n$; this is also true of $M(a,b,z)$ provided that $b$ is not a nonpositive integer. … ►When $a=-m$, $m=0,1,2,\mathrm{\dots }$, $U(a,b,z)$ is a polynomial in $z$ of degree $m$: … ►When $m\in \mathbb{Z}$, … ►Also, when $b$ is not an integer …

36: Frank W. J. Olver

… ►degrees in mathematics from the University of London in 1945, 1948, and 1961, respectively. … ►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. …

37: Philip J. Davis

… ► 2018) received an undergraduate degree in mathematics from Harvard College in 1943. … ►Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958. …

38: Bibliography C

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An algorithm for exponential integrals of real order. Computing 45 (3), pp. 269–276.

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Notes:

Includes Fortran code

External Links:

ISSN 0010-485X, Document, MathReview, ZentralBlatt

Referenced by:

§8.25(v), §8.28(vi).

Encodings:

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The zeros of Hankel functions as functions of their order. Numer. Math. 7 (3), pp. 238–250.

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External Links:

ISSN 0029-599X, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(xiv).

Encodings:

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Derivatives with respect to the degree and order of associated Legendre functions for $|z|>1$ using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Dmitriy Karp), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

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Chazy’s second-degree Painlevé equations. J. Phys. A 39 (39), pp. 11955–11971.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§32.7(vii).

Encodings:

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Zeros of the Hankel function of real order out of the principal Riemann sheet. J. Comput. Appl. Math. 37 (1-3), pp. 89–99.

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External Links:

ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

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39: 18.15 Asymptotic Approximations

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18.15.1 $${\left(\sin \frac{1}{2}\theta \right)}^{\alpha +\frac{1}{2}}{\left(\cos \frac{1}{2}\theta \right)}^{\beta +\frac{1}{2}}{P}_n^{(\alpha ,\beta )}\left(\cos \theta \right)={\pi }^{-1}{2}^{2n+\alpha +\beta +1}\mathrm{B}(n+\alpha +1,n+\beta +1)\left(\sum _{m=0}^{M-1}\frac{f_m(\theta )}{2^m{\left(2n+\alpha +\beta +2\right)}_m}+O\left(n^{-M}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{B}(a,b)$: beta function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer and $f_m(\theta )$

Referenced by:

§18.15(i), §18.15(i)

Permalink:

http://dlmf.nist.gov/18.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

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18.15.12 $$P_n\left(\cos \theta \right)={\left(\frac{2}{\sin \theta }\right)}^{\frac{1}{2}}\sum _{m=0}^{M-1}\left(\genfrac{}{}{0.0pt}{}{-\frac{1}{2}}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m-\frac{1}{2}}{n}\right)\frac{\cos {\alpha }_{n,m}}{{(2\sin \theta )}^m}+O\left(\frac{1}{n^{M+\frac{1}{2}}}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $P_n\left(x\right)$: Legendre polynomial, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\cos z$: cosine function, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer and ${\alpha }_{n,m}$

Referenced by:

§18.15(iii)

Permalink:

http://dlmf.nist.gov/18.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(iii), §18.15 and Ch.18

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18.15.22 $$L_n^{(\alpha )}\left(\nu x\right)={(-1)}^n\frac{{\mathrm{e}}^{\frac{1}{2}\nu x}}{2^{\alpha -\frac{1}{2}}{x}^{\frac{1}{2}\alpha +\frac{1}{4}}}{\left(\frac{\zeta }{x-1}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{m=0}^{M-1}\frac{E_m(\zeta )}{{\nu }^{2m}}+\frac{{\mathrm{Ai}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{m=0}^{M-1}\frac{F_m(\zeta )}{{\nu }^{2m}}+\mathrm{envAi}\left({\nu }^{\frac{2}{3}}\zeta \right)O\left(\frac{1}{{\nu }^{2M-\frac{2}{3}}}\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\sim$: Poincaré asymptotic expansion, $\mathrm{envAi}\left(x\right)$: envelope of Airy function $\mathrm{Ai}\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $m$: nonnegative integer, $n$: nonnegative integer, $\nu$ , $\zeta$ , $E_m$: coefficient, $F_m$: coefficient and $x$: real variable

Referenced by:

Erratum (V1.0.11) for Equation (18.15.22)

Permalink:

http://dlmf.nist.gov/18.15.E22

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally this equation was expressed in terms of the asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right-hand side, $\sim$ was replaced by $=$.

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

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18.15.27 $$H_n\left(x\right)={\lambda }_n{\mathrm{e}}^{\frac{1}{2}{x}^2}\left(\sum _{m=0}^{M-1}\frac{u_m(x)\cos {\omega }_{n,m}(x)}{{\mu }^{\frac{1}{2}m}}+O\left(\frac{1}{{\mu }^{\frac{1}{2}M}}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $H_n\left(x\right)$: Hermite polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $m$: nonnegative integer, $n$: nonnegative integer, ${\omega }_{n,m}(x)$ and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.15.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(v), §18.15 and Ch.18

… ►The asymptotic behavior of the classical OP’s as $x\to \pm \mathrm{\infty }$ with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. …

40: 14.17 Integrals

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14.17.1 $$\int {\left(1-x^2\right)}^{-\mu /2}{𝖯}_{\nu }^{\mu }\left(x\right)dx=-{\left(1-x^2\right)}^{-(\mu -1)/2}{𝖯}_{\nu }^{\mu -1}\left(x\right).$$

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Symbols:

: Ferrers function of the first kind, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.17(i)

Permalink:

http://dlmf.nist.gov/14.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(i), §14.17 and Ch.14

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14.17.2 $$\int {\left(1-x^2\right)}^{\mu /2}{𝖯}_{\nu }^{\mu }\left(x\right)dx=\frac{{\left(1-x^2\right)}^{(\mu +1)/2}}{(\nu -\mu )(\nu +\mu +1)}{𝖯}_{\nu }^{\mu +1}\left(x\right),$$ $\mu \ne \nu$ or $-\nu -1$.

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Symbols:

: Ferrers function of the first kind, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(i), §14.17 and Ch.14

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14.17.5 $${\int }_0^1{x}^{\sigma }{\left(1-x^2\right)}^{\mu /2}{𝖯}_{\nu }^{-\mu }\left(x\right)dx=\frac{\mathrm{\Gamma }\left(\frac{1}{2}\sigma +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\sigma +1\right)}{2^{\mu +1}\mathrm{\Gamma }\left(\frac{1}{2}\sigma -\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)\mathrm{\Gamma }\left(\frac{1}{2}\sigma +\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{2}\right)},$$ $\mathrm{\Re }\sigma >-1$, $\mathrm{\Re }\mu >-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(ii), §14.17 and Ch.14

… ►For $l,m,n=0,1,2,\mathrm{\dots }$, … ►Orthogonality relations for the associated Legendre functions of imaginary order are given in Bielski (2013). …