limit relations

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11: 18.34 Bessel Polynomials

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18.34.8 $$\underset{\alpha \to \mathrm{\infty }}{lim}\frac{P_n^{(\alpha ,a-\alpha -2)}\left(1+\alpha x\right)}{P_n^{(\alpha ,a-\alpha -2)}\left(1\right)}=y_n(x;a).$$

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Symbols:

$y_n(x;a)$: Bessel polynomial, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.34(iii)

Permalink:

http://dlmf.nist.gov/18.34.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.34(iii), §18.34 and Ch.18

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12: Bibliography R

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Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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External Links:

ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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13: 1.9 Calculus of a Complex Variable

… ►Also, the union of $S$ and its limit points is the closure of $S$. … ►A function $f(z)$ is complex differentiable at a point $z$ if the following limit exists: … ►or its limiting form, and is invariant under bilinear transformations. … ►

§1.9(vii) Inversion of Limits

… ►Then both repeated limits equal $z$. …

14: 31.9 Orthogonality

… ►The right-hand side may be evaluated at any convenient value, or limiting value, of $\zeta$ in $(0,1)$ since it is independent of $\zeta$. ►For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64). … ►For bi-orthogonal relations for path-multiplicative solutions see Schmidt (1979, §2.2). …

15: 11.10 Anger–Weber Functions

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§11.10(vi) Relations to Other Functions

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11.10.18 $${𝐄}_{\nu }\left(z\right)=-\frac{1}{\pi }(1+\cos \left(\pi \nu \right))s_{0,\nu }\left(z\right)-\frac{\nu }{\pi }(1-\cos \left(\pi \nu \right))s_{-1,\nu }\left(z\right).$$

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Symbols:

$s_{\mu ,\nu }\left(z\right)$: Lommel function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vi)

Permalink:

http://dlmf.nist.gov/11.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vi), §11.10 and Ch.11

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$m_2=\lceil \frac{1}{2}n-\frac{3}{2}\rceil .$

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§11.10(ix) Recurrence Relations and Derivatives

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16: 18.22 Hahn Class: Recurrence Relations and Differences

§18.22 Hahn Class: Recurrence Relations and Differences

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§18.22(i) Recurrence Relations in $n$

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Table 18.22.1: Recurrence relations (18.22.2) for Krawtchouk, Meixner, and Charlier polynomials.

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$p_n(x)$	$A_n$	$C_n$
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17: 8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

… ►The right-hand sides are replaced by their limiting forms when $p=1,2,3,\mathrm{\dots }$. ►

§8.19(v) Recurrence Relation and Derivatives

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§8.19(vi) Relation to Confluent Hypergeometric Function

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18: 18.23 Hahn Class: Generating Functions

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Hahn

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19: 7.2 Definitions

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$\underset{z\to \mathrm{\infty }}{lim}\mathrm{erf}z=1,$

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$\underset{z\to \mathrm{\infty }}{lim}\mathrm{erfc}z=0$ , .

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$\underset{x\to \mathrm{\infty }}{lim}C\left(x\right)=\frac{1}{2},$

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$\underset{x\to \mathrm{\infty }}{lim}S\left(x\right)=\frac{1}{2}.$

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§7.2(iv) Auxiliary Functions

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20: 18.2 General Orthogonal Polynomials

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§18.2(iii) Standardization and Related Constants

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§18.2(iv) Recurrence Relations

… ►the monic recurrence relations (18.2.8) and (18.2.10) take the form … ►The recurrence relations (18.2.10) can be equivalently written as … ►are OP’s with orthogonality relation …