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1: 26.12 Plane Partitions

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26.12.17 $$\prod _{h=0}^{r-1}\frac{(3h+1)!}{(r+h)!}.$$

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Symbols:

$!$: factorial (as in $n!$), $r$: positive integer and $h$: positive integer

Permalink:

http://dlmf.nist.gov/26.12.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(i), §26.12 and Ch.26

►A strict shifted plane partition is an arrangement of the parts in a partition so that each row is indented one space from the previous row and there is weak decrease across rows and strict decrease down columns. … ►A descending plane partition is a strict shifted plane partition in which the number of parts in each row is strictly less than the largest part in that row and is greater than or equal to the largest part in the next row. The example of a strict shifted plane partition also satisfies the conditions of a descending plane partition. …

2: 33.25 Approximations

§33.25 Approximations

►Cody and Hillstrom (1970) provides rational approximations of the phase shift ${\sigma }_0\left(\eta \right)=\mathrm{ph}\mathrm{\Gamma }\left(1+\mathrm{i}\eta \right)$ (see (33.2.10)) for the ranges $0\le \eta \le 2$, $2\le \eta \le 4$, and $4\le \eta \le \mathrm{\infty }$. …

3: 26.10 Integer Partitions: Other Restrictions

… ►Note that $p({𝒟}^{\prime }3,n)\le p(𝒟3,n)$, with strict inequality for $n\ge 9$. It is known that for $k>3$, $p(𝒟k,n)\ge p(\in A_{1,k+3},n)$, with strict inequality for $n$ sufficiently large, provided that $k=2^m-1,m=3,4,5$, or $k\ge 32$; see Yee (2004). …

4: Possible Errors in DLMF

… ►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula. …

5: 6.15 Sums

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6.15.2 $$\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{si}\left(\pi n\right)}{n}=\frac{1}{2}\pi (\ln \pi -1),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

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6.15.4 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{\mathrm{si}\left(2\pi n\right)}{n}=\pi (\frac{3}{2}\ln 2-1).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

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6: 33.13 Complex Variable and Parameters

… ►The quantities $C_{\mathrm{\ell }}\left(\eta \right)$, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$, and $R_{\mathrm{\ell }}$, given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that ►

33.13.1 $$C_{\mathrm{\ell }}\left(\eta \right)=2^{\mathrm{\ell }}{\mathrm{e}}^{\mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)-(\pi \eta /2)}\mathrm{\Gamma }\left(\mathrm{\ell }+1-\mathrm{i}\eta \right)/\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right),$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

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7: 6.14 Integrals

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6.14.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{si}\left(t\right)dt=-\frac{1}{a}\arctan a,$$ $\mathrm{\Re }a>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.29

Permalink:

http://dlmf.nist.gov/6.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

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6.14.5 $${\int }_0^{\mathrm{\infty }}\cos t\mathrm{Ci}\left(t\right)dt={\int }_0^{\mathrm{\infty }}\sin t\mathrm{si}\left(t\right)dt=-\frac{1}{4}\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\int$: integral, $\mathrm{si}\left(z\right)$: sine integral and $\sin z$: sine function

A&S Ref:

5.2.30

Permalink:

http://dlmf.nist.gov/6.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

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6.14.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{Ci}}^2\left(t\right)dt={\int }_0^{\mathrm{\infty }}{\mathrm{si}}^2\left(t\right)dt=\frac{1}{2}\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\int$: integral and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.31

Permalink:

http://dlmf.nist.gov/6.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

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6.14.7 $${\int }_0^{\mathrm{\infty }}\mathrm{Ci}\left(t\right)\mathrm{si}\left(t\right)dt=\ln 2.$$

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Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.32

Referenced by:

§6.14(ii)

Permalink:

http://dlmf.nist.gov/6.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

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8: 18.3 Definitions

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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
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Shifted Chebyshev of first kind	$T_n^{\ast }\left(x\right)$	$(0,1)$	${(x-x^2)}^{-\frac{1}{2}}$			$-\frac{1}{2}n$
Shifted Chebyshev of second kind	$U_n^{\ast }\left(x\right)$	$(0,1)$	${(x-x^2)}^{\frac{1}{2}}$	$\frac{1}{8}\pi$	$2^{2n}$	$-\frac{1}{2}n$
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Shifted Legendre	$P_n^{\ast }\left(x\right)$	$(0,1)$	$1$	$1/(2n+1)$	$2^{2n}{\left(\frac{1}{2}\right)}_n/n!$	$-\frac{1}{2}n$
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9: 35.4 Partitions and Zonal Polynomials

… ►Also, $|\kappa |$ denotes $k_1+\mathrm{\cdots }+k_m$, the weight of $\kappa$; $\mathrm{\ell }(\kappa )$ denotes the number of nonzero $k_j$; $a+\kappa$ denotes the vector $(a+k_1,\mathrm{\dots },a+k_m)$. ►The partitional shifted factorial is given by ►

35.4.1 $${\left[a\right]}_{\kappa }=\frac{{\mathrm{\Gamma }}_m\left(a+\kappa \right)}{{\mathrm{\Gamma }}_m\left(a\right)}=\prod _{j=1}^m{\left(a-\frac{1}{2}(j-1)\right)}_{k_j},$$

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Defines:

${\left[a\right]}_{\kappa }$: partitional shifted factorial

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

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35.4.2

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Symbols:

$!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer, $\kappa$: vector of nonnegative integers and $\mathrm{\ell }(\kappa )$: number of nonzeros

Referenced by:

§35.4(i)

Permalink:

http://dlmf.nist.gov/35.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

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35.4.9

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Symbols:

$\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer, $𝟎$: $m\times m$ matrix of zeros and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

10: 18.1 Notation

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Classical OP’s

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Shifted Chebyshev of first and second kinds: $T_n^{\ast }\left(x\right)$, $U_n^{\ast }\left(x\right)$.

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Shifted Legendre: $P_n^{\ast }\left(x\right)$.

… ►Nor do we consider the shifted Jacobi polynomials: ►

18.1.2 $$G_n(p,q,x)=\frac{n!}{{\left(n+p\right)}_n}{P}_n^{(p-q,q-1)}\left(2x-1\right),$$

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Defines:

$G_n(p,q,x)$: shifted Jacobi polynomial

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $q$: real variable, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.1.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.1(iii), §18.1 and Ch.18

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