q-multinomial%20coefficient

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1: 26.16 Multiset Permutations

… ►Thus

\mathop{\mathrm{inv}}(351322453154)=4+8+0+3+1+1+2+3+1+0+1=24

, and

\mathop{\mathrm{maj}}(351322453154)=2+4+8+9+11=34.

►The $q$ -multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by ►

26.16.1

\genfrac{[}{]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{q% }=\prod_{k=1}^{n-1}\genfrac{[}{]}{0.0pt}{}{a_{k}+a_{k+1}+\cdots+a_{n}}{a_{k}}_% {q},

ⓘ

Symbols:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $\genfrac{[}{]}{0.0pt}{}{\NVar{a_{1}}+\NVar{a_{2}}+\dots+\NVar{a_{n}}}{\NVar{a_% {1}},\NVar{a_{2}},\ldots,\NVar{a_{n}}}_{\NVar{q}}$ : $q$ -multinomial coefficient, $k$ : nonnegative integer, $n$ : nonnegative integer, $a_{j}$ : numbers and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.16 and Ch.26

… ►

26.16.2

\sum_{\sigma\in\mathfrak{S}_{S}}q^{\mathop{\mathrm{inv}}(\sigma)}=\genfrac{[}{% ]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{q},

ⓘ

Symbols:: $\in$ : element of, $\genfrac{[}{]}{0.0pt}{}{\NVar{a_{1}}+\NVar{a_{2}}+\dots+\NVar{a_{n}}}{\NVar{a_% {1}},\NVar{a_{2}},\ldots,\NVar{a_{n}}}_{\NVar{q}}$ : $q$ -multinomial coefficient, $\mathfrak{S}_{\NVar{n}}$ : set of permutations of $\{1,2,\ldots,n\}$ , $n$ : nonnegative integer, $a_{j}$ : numbers, $S$ : multiset and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §26.16 and Ch.26

►

26.16.3

\sum_{\sigma\in\mathfrak{S}_{S}}q^{\mathop{\mathrm{maj}}(\sigma)}=\genfrac{[}{% ]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{q}.

ⓘ

Symbols:: $\in$ : element of, $\genfrac{[}{]}{0.0pt}{}{\NVar{a_{1}}+\NVar{a_{2}}+\dots+\NVar{a_{n}}}{\NVar{a_% {1}},\NVar{a_{2}},\ldots,\NVar{a_{n}}}_{\NVar{q}}$ : $q$ -multinomial coefficient, $\mathfrak{S}_{\NVar{n}}$ : set of permutations of $\{1,2,\ldots,n\}$ , $n$ : nonnegative integer, $a_{j}$ : numbers, $S$ : multiset and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.16.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §26.16 and Ch.26

2: 26.3 Lattice Paths: Binomial Coefficients

§26.3 Lattice Paths: Binomial Coefficients

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§26.3(i) Definitions

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§26.3(ii) Generating Functions

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§26.3(iii) Recurrence Relations

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§26.3(iv) Identities

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4: 7.24 Approximations

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•

Cody (1969) provides minimax rational approximations for $\operatorname{erf}x$ and $\operatorname{erfc}x$ . The maximum relative precision is about 20S.

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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Luke (1969b, pp. 323–324) covers $\frac{1}{2}\sqrt{\pi}\operatorname{erf}x$ and $e^{x^{2}}F\left(x\right)$ for $-3\leq x\leq 3$ (the Chebyshev coefficients are given to 20D); $\sqrt{\pi}xe^{x^{2}}\operatorname{erfc}x$ and $2xF\left(x\right)$ for $x\geq 3$ (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

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Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ for $x\geq 3$ (15D).

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Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x)e^{x^{2}}\operatorname{erfc}x$ on $(0,\infty)$ (22D).

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5: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

§26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

►

§26.4(i) Definitions

… ►

M_{1}

is the multinominal coefficient (26.4.2): … ►

§26.4(ii) Generating Function

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§26.4(iii) Recurrence Relation

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6: 6.20 Approximations

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Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$ , with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\operatorname{Ei}\left(x\right)$ , with accuracies up to 20S.

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$ , with accuracies up to 20S.

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Clenshaw (1962) gives Chebyshev coefficients for $-E_{1}\left(x\right)-\ln|x|$ for $-4\leq x\leq 4$ and $e^{x}E_{1}\left(x\right)$ for $x\geq 4$ (20D).

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Luke (1969b, pp. 321–322) covers $\operatorname{Ein}\left(x\right)$ and $-\operatorname{Ein}\left(-x\right)$ for $0\leq x\leq 8$ (the Chebyshev coefficients are given to 20D); $E_{1}\left(x\right)$ for $x\geq 5$ (20D), and $\operatorname{Ei}\left(x\right)$ for $x\geq 8$ (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

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7: 25.20 Approximations

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Cody et al. (1971) gives rational approximations for $\zeta\left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\leq s\leq 5$ , $5\leq s\leq 11$ , $11\leq s\leq 25$ , $25\leq s\leq 55$ . Precision is varied, with a maximum of 20S.

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Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta\left(s+1\right)$ and $\zeta\left(s+k\right)$ , $k=2,3,4,5,8$ , for $0\leq s\leq 1$ (23D).

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Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover $\zeta\left(s\right)$ for $0\leq s\leq 1$ (15D), $\zeta\left(s+1\right)$ for $0\leq s\leq 1$ (20D), and $\ln\xi\left(\tfrac{1}{2}+ix\right)$ (§25.4) for $-1\leq x\leq 1$ (20D). For errata see Piessens and Branders (1972).

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8: 28.35 Tables

§28.35 Tables

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•

Ince (1932) includes eigenvalues $a_{n}$ , $b_{n}$ , and Fourier coefficients for $n=0$ or $1(1)6$ , $q=0(1)10(2)20(4)40$ ; 7D. Also $\operatorname{ce}_{n}\left(x,q\right)$ , $\operatorname{se}_{n}\left(x,q\right)$ for $q=0(1)10$ , $x=1(1)90$ , corresponding to the eigenvalues in the tables; 5D. Notation: $a_{n}=\mathit{be}_{n}-2q$ , $b_{n}=\mathit{bo}_{n}-2q$ .

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Kirkpatrick (1960) contains tables of the modified functions $\operatorname{Ce}_{n}\left(x,q\right)$ , $\operatorname{Se}_{n+1}\left(x,q\right)$ for $n=0(1)5$ , $q=1(1)20$ , $x=0.1(.1)1$ ; 4D or 5D.

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National Bureau of Standards (1967) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n}\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$ , and $n=4(1)15$ with $q=0(2)100$ ; Fourier coefficients for $\operatorname{ce}_{n}\left(x,q\right)$ and $\operatorname{se}_{n}\left(x,q\right)$ for $n=0(1)15$ , $n=1(1)15$ , respectively, and various values of $q$ in the interval $[0,100]$ ; joining factors $g_{\mathit{e},n}(\sqrt{q})$ , $f_{\mathit{e},n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\mbox{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$ . Precision is generally 8D.

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Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n+1}\left(q\right)$ for $n=0(1)4$ , $q=0(1)50$ ; $n=0(1)20$ ( $a$ ’s) or 19 ( $b$ ’s), $q=1,3,5,10,15,25,50(50)200$ . Fourier coefficients for $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , $n=0(1)7$ . Mathieu functions $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , and their first $x$ -derivatives for $n=0(1)4$ , $x=0(5^{\circ})90^{\circ}$ . Modified Mathieu functions ${\operatorname{Mc}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ , ${\operatorname{Ms}^{(j)}_{n+1}}\left(x,\sqrt{10}\right)$ , and their first $x$ -derivatives for $n=0(1)4$ , $j=1,2$ , $x=0(.2)4$ . Precision is mostly 9S.

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9: 10.75 Tables

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Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

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Olver (1960) tabulates $j_{n,m}$ , $J_{n}'\left(j_{n,m}\right)$ , ${j^{\prime}_{n,m}}$ , $J_{n}\left({j^{\prime}_{n,m}}\right)$ , $y_{n,m}$ , $Y_{n}'\left(y_{n,m}\right)$ , ${y^{\prime}_{n,m}}$ , $Y_{n}\left({y^{\prime}_{n,m}}\right)$ , $n=0(\tfrac{1}{2})20\tfrac{1}{2}$ , $m=1(1)50$ , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ ; see §10.21(viii), and more fully Olver (1954).

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Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

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Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_{n}\left(z\right)$ and $K_{n}'\left(z\right)$ , for $n=2(1)20$ , 9S.

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Olver (1960) tabulates $a^{\prime}_{n,m}$ , $\mathsf{j}_{n}\left(a^{\prime}_{n,m}\right)$ , $b^{\prime}_{n,m}$ , $\mathsf{y}_{n}\left(b^{\prime}_{n,m}\right)$ , $n=1(1)20$ , $m=1(1)50$ , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ .

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10: 3.4 Differentiation

… ►The

B_{k}^{n}

are the differentiated Lagrangian interpolation coefficients: … ►

B_{-2}^{5}=\tfrac{1}{120}(6-10t-15t^{2}+20t^{3}-5t^{4}),

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B_{-3}^{6}=-\tfrac{1}{720}(12-8t-45t^{2}+20t^{3}+15t^{4}-6t^{5}),

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B_{-2}^{6}=\tfrac{1}{60}(9-9t-30t^{2}+20t^{3}+5t^{4}-3t^{5}),

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B_{2}^{6}=-\tfrac{1}{60}(9+9t-30t^{2}-20t^{3}+5t^{4}+3t^{5}),

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