convergence properties

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31: 30.4 Functions of the First Kind

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§30.4(ii) Elementary Properties

… ►The expansion (30.4.7) converges in the norm of $L^2(-1,1)$, that is, …It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for $-1\le x\le 1$. …

32: 5.11 Asymptotic Expansions

… ►The scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ is defined in (5.11.3) and its main property is ${\mathrm{\Gamma }}^{\ast }\left(z\right)\sim 1$ as $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$. … ►For similar results including a convergent factorial series see, Nemes (2013c). …

33: 16.4 Argument Unity

… ►The function ${}_3{}^{}F_2^{}(a,b,c;d,e;1)$ is analytic in the parameters $a,b,c,d,e$ when its series expansion converges and the bottom parameters are not negative integers or zero. … ►The characterizing properties (18.22.2), (18.22.10), (18.22.19), (18.22.20), and (18.26.14) of the Hahn and Wilson class polynomials are examples of the contiguous relations mentioned in the previous three paragraphs. … ►when the series on the right terminates and the series on the left converges. …

34: Bibliography S

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A property of the zeros of cross-product Bessel functions of different orders. Z. Angew. Math. Mech. 56 (2), pp. 120–121.

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Document, MathReview (S. Saran), ZentralBlatt

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§10.21(x).

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Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

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MathReview (J. Kuntzmann), ZentralBlatt

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§17.18.

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Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.

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ISSN 0012-7094, Link, MathReview (A. Erdélyi), ZentralBlatt

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§18.2(xii).

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Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.

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ISSN 0285-4821, MathReview (Peter M. Neumann), ZentralBlatt

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§24.10(ii).

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Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 (1), pp. 76–115.

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ISSN 0001-8708, Document, MathReview (Dennis White), ZentralBlatt

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§18.37(iii).

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35: 1.17 Integral and Series Representations of the Dirac Delta

… ►This is a symbolic function with the properties: … ►for all functions $\varphi (x)$ that are continuous when $x\in (-\mathrm{\infty },\mathrm{\infty })$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. … ►More generally, assume $\varphi (x)$ is piecewise continuous (§1.4(ii)) when $x\in [-c,c]$ for any finite positive real value of $c$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. … ►The inner integral does not converge. … ►The sum ${\sum }_{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}k(x-a)}$ does not converge, but (1.17.18) can be interpreted as a generalized integral in the sense that …

36: Bibliography G

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On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.

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Orthogonal polynomials and numerical methods

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ISSN 0377-0427, Document, Link, MathReview (Mircea Ivan), ZentralBlatt

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Erratum (V1.0.28) for Table 18.3.1.

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

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

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MathReview, ZentralBlatt

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§2.6(i).

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Generalized Fermi-Dirac functions and derivatives: Properties and evaluation. Comput. Phys. Comm. 136 (3), pp. 294–309.

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Double-precision Fortran, maximum accuracy 20D

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Document, Code, ZentralBlatt

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7th item.

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Do integrable mappings have the Painlevé property?. Phys. Rev. Lett. 67 (14), pp. 1825–1828.

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ISSN 0031-9007, Document, MathReview, ZentralBlatt

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§32.16.

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A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.

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ISSN 0019-3577, Document, MathReview (Sumitra Purkayastha), ZentralBlatt

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§35.9.

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37: Mathematical Introduction

… ►In addition, there is a comprehensive account of the great variety of analytical methods that are used for deriving and applying the extremely important asymptotic properties of the special functions, including double asymptotic properties (Chapter 2 and §§10.41(iv), 10.41(v)). … ► ►►►

$ℂ$	complex plane (excluding infinity).
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	is finite, or converges.
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38: 27.14 Unrestricted Partitions

… ►Rademacher (1938) derives a convergent series that also provides an asymptotic expansion for $p\left(n\right)$: … ►For further properties of the function $\eta \left(\tau \right)$ see §§23.15–23.19. ►

§27.14(v) Divisibility Properties

►Ramanujan (1921) gives identities that imply divisibility properties of the partition function. …

39: 18.2 General Orthogonal Polynomials

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Kernel property

… ►For illustrations of these properties see Figures 18.4.1–18.4.7. … ►(convergence in $L_w^2((a,b))$). … ►for $x,y$ in the support of the orthogonality measure and $z$ such that the series in (18.2.41) converges absolutely for all these $x,y$. … ►If $v(s)$ is the formal power series such that $v(u(t))=t$ then a property equivalent to (18.2.45) with $c_n=1$ is that …

40: 19.16 Definitions

… ►In (19.16.1)–(19.16.2_5), $x,y,z\in ℂ\setminus (-\mathrm{\infty },0]$ except that one or more of $x,y,z$ may be 0 when the corresponding integral converges. … ►The $R$-function is often used to make a unified statement of a property of several elliptic integrals. … ►When one variable is 0 without destroying convergence, any one of (19.16.14)–(19.16.17) is said to be complete and can be written as an $R$-function with one less variable: …