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11: 12.12 Integrals

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Nicholson-type Integral

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12: How to Cite

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Item Type	Ref. Number	Permalink
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13: Foreword

… ►The provision of standard reference data of this type is a core function of NIST. …

14: 10.44 Sums

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§10.44(iii) Neumann-Type Expansions

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15: 31.11 Expansions in Series of Hypergeometric Functions

… ►The series of Type I (§31.11(iii)) are useful since they represent the functions in large domains. … ►

§31.11(iii) Type I

… ►Every Fuchs–Frobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. … ►

§31.11(iv) Type II

… ►Every Heun function can be represented by a series of Type II. …

16: 7.24 Approximations

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§7.24(iii) Padé-Type Expansions

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17: 18.14 Inequalities

… ►Equations (18.14.3) and (18.14.3_5) are Bernstein-type inequalities. … … ►

§18.14(ii) Turán-Type Inequalities

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Legendre

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Jacobi

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18: 18.27 $q$-Hahn Class

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18.27.6 $$P_n^{(\alpha ,\beta )}(x;c,d;q)=\frac{c^n{q}^{-(\alpha +1)n}{(q^{\alpha +1},-q^{\alpha +1}{c}^{-1}d;q)}_n}{{(q,-q;q)}_n}{P}_n(q^{\alpha +1}{c}^{-1}x;q^{\alpha },q^{\beta },-q^{\alpha }{c}^{-1}d;q),$$

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Defines:

$P_n^{(\alpha ,\beta )}(x;c,d;q)$: big $q$-Jacobi polynomial and $P_n^{(\alpha ,\beta )}(x;c,d;q)$ (locally)

Symbols:

$(a,b)$: open interval, $P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Andrews and Askey (1985, (3.28))

Referenced by:

(18.27.12_5), §18.27(iii), Erratum (V1.0.17) for Equation (18.27.6)

Permalink:

http://dlmf.nist.gov/18.27.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.17):

Originally the first argument of the big $q$-Jacobi polynomial on the right-hand side was written incorrectly as $q^{\alpha +1}{c}^{-1}dx$.

Reported 2017-09-27 by Tom Koornwinder

See also:

Annotations for §18.27(iii), §18.27 and Ch.18

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18.27.10 $$p_n(x)=P_n^{(\alpha ,\beta )}(x;c,d;q)$$

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Symbols:

$P_n^{(\alpha ,\beta )}(x;c,d;q)$: big $q$-Jacobi polynomial, $p_n(x)$: polynomial of degree $n$ , $q$: real variable, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.27.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.27(iii), §18.27 and Ch.18

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18.27.12_5 $$\underset{q\to 1}{lim}{P}_n^{(\alpha ,\beta )}(x;c,d;q)={\left(\frac{c+d}{2}\right)}^n{P}_n^{(\alpha ,\beta )}\left(\frac{2x-c+d}{c+d}\right).$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $P_n^{(\alpha ,\beta )}(x;c,d;q)$: big $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Andrews and Askey (1985, (3.33)); for $c=d=1$

Proof sketch:

On the left substitute first (18.27.6) and next (18.27.5). Then, after taking the limit, a ${}_2{}^{}F_1^{}$ hypergeometric function occurs. Finally use (18.5.7).

Referenced by:

§18.27(iii), §18.27(iii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E12_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(iii), §18.27(iii), §18.27 and Ch.18

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19: 26.13 Permutations: Cycle Notation

… ►An element of ${𝔖}_n$ with $a_1$ fixed points, $a_2$ cycles of length $2,\mathrm{\dots },a_n$ cycles of length $n$, where $n=a_1+2a_2+\mathrm{\cdots }+n{a}_n$, is said to have cycle type $\left(a_1,a_2,\mathrm{\dots },a_n\right)$. The number of elements of ${𝔖}_n$ with cycle type $\left(a_1,a_2,\mathrm{\dots },a_n\right)$ is given by (26.4.7). … ►A permutation with cycle type $\left(a_1,a_2,\mathrm{\dots },a_n\right)$ can be written as a product of $a_2+2a_3+\mathrm{\cdots }+(n-1)a_n=n-(a_1+a_2+\mathrm{\cdots }+a_n)$ transpositions, and no fewer. …

20: Bille C. Carlson

… ►Also, the homogeneity of the $R$-function has led to a new type of mean value for several variables, accompanied by various inequalities. …