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1: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

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2: William P. Reinhardt

… ►Reinhardt is a frequent visitor to the NIST Physics Laboratory in Gaithersburg, and to the Joint Quantum Institute (JQI) and Institute for Physical Sciences and Technology (ISTP) at the University of Maryland. … ►He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions. Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. … ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

3: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

… ►However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of $a$, $b$, and $x$, and also to complex values. … ►

§8.17(ii) Hypergeometric Representations

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§8.17(iv) Recurrence Relations

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§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

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4: Gergő Nemes

… ►Nemes has research interests in asymptotic analysis, Écalle theory, exact WKB analysis, and special functions. ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. ►In March 2022, Nemes was named Contributing Developer of the NIST Digital Library of Mathematical Functions.

5: Wolter Groenevelt

… ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. ►In July 2023, Groenevelt was named Contributing Developer of the NIST Digital Library of Mathematical Functions.

6: Bibliography F

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

Encodings:

BibTeX

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Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:

Fortran, minimum precision 6D.

External Links:

ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(ix).

Encodings:

BibTeX

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

Encodings:

BibTeX

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7: 26.3 Lattice Paths: Binomial Coefficients

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§26.3(i) Definitions

► $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ is the number of ways of choosing $n$ objects from a collection of $m$ distinct objects without regard to order. $\left(\genfrac{}{}{0.0pt}{}{m+n}{n}\right)$ is the number of lattice paths from $(0,0)$ to $(m,n)$. …The number of lattice paths from $(0,0)$ to $(m,n)$, $m\le n$, that stay on or above the line $y=x$ is $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)-\left(\genfrac{}{}{0.0pt}{}{m+n}{m-1}\right).$ … ►

§26.3(iii) Recurrence Relations

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8: 26.5 Lattice Paths: Catalan Numbers

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§26.5(i) Definitions

… ►It counts the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$. … ►

§26.5(ii) Generating Function

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§26.5(iii) Recurrence Relations

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26.5.6 $$C\left(n\right)\sim \frac{4^n}{\sqrt{\pi n^3}},$$ $n\to \mathrm{\infty }$,

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Symbols:

$C\left(n\right)$: Catalan number, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iv), §26.5 and Ch.26

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9: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

… ►The Riemann zeta function is a special case: … ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►When $a=1$, (25.11.35) reduces to (25.2.3). … ►uniformly with respect to bounded nonnegative values of $\alpha$. …

10: 33.24 Tables

§33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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Curtis (1964a) tabulates $P_{\mathrm{\ell }}(ϵ,r)$, $Q_{\mathrm{\ell }}(ϵ,r)$ (§33.1), and related functions for $\mathrm{\ell }=0,1,2$ and $ϵ=-2(.2)2$, with $x=0(.1)4$ for and $x=0(.1)10$ for $ϵ\ge 0$; 6D.

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