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1: 19.2 Definitions
where $p_{j}$ is a polynomial in $t$ while $\rho$ and $\sigma$ are rational functions of $t$. … Here $a,b,p$ are real parameters, and $k_{c}$ and $x$ are real or complex variables, with $p\neq 0$, $k_{c}\neq 0$. … If $1, then $k_{c}$ is pure imaginary. …
§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$
For the special cases of $R_{C}\left(x,x\right)$ and $R_{C}\left(0,y\right)$ see (19.6.15). …
2: 34.6 Definition: $\mathit{9j}$ Symbol
34.6.1 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},$
34.6.2 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.$
3: 26.9 Integer Partitions: Restricted Number and Part Size
$p_{k}\left(n\right)$ denotes the number of partitions of $n$ into at most $k$ parts. See Table 26.9.1. … It follows that $p_{k}\left(n\right)$ also equals the number of partitions of $n$ into parts that are less than or equal to $k$. $p_{k}\left(\leq m,n\right)$ is the number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$. …
4: 34.7 Basic Properties: $\mathit{9j}$ Symbol
34.7.1 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{13}\\ j_{31}&j_{31}&0\end{Bmatrix}=\frac{(-1)^{j_{12}+j_{21}+j_{13}+j_{31}}}{((2j_{1% 3}+1)(2j_{31}+1))^{\frac{1}{2}}}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{22}&j_{21}&j_{31}\end{Bmatrix}.$
34.7.2 $\sum_{j_{12}\,j_{34}}(2j_{12}+1)(2j_{34}+1)(2j_{13}+1)(2j_{24}+1)\begin{% Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j^{\prime}_{13}&j^{\prime}_{24}&j\end{Bmatrix}=\delta_{j_{13},j^{\prime}_{13}}% \delta_{j_{24},j^{\prime}_{24}}.$
34.7.3 $\sum_{j_{13}\,j_{24}}(-1)^{2j_{2}+j_{24}+j_{23}-j_{34}}(2j_{13}+1)(2j_{24}+1)% \begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{3}&j_{13}\\ j_{4}&j_{2}&j_{24}\\ j_{14}&j_{23}&j\end{Bmatrix}=\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{4}&j_{3}&j_{34}\\ j_{14}&j_{23}&j\end{Bmatrix}.$
34.7.4 $\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{m_{r1},m_{r2},r=1,2,3}\begin{pmatrix}j% _{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\*\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}.$
34.7.5 $\sum_{j^{\prime}}(2j^{\prime}+1)\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{23}&j_{33}&j\end{Bmatrix}={(-1)^{2j}}\begin{Bmatrix}j_{21}&j_{22}&j_{23}\\ j_{12}&j&j_{32}\end{Bmatrix}\begin{Bmatrix}j_{31}&j_{32}&j_{33}\\ j&j_{11}&j_{21}\end{Bmatrix}.$
5: 28.6 Expansions for Small $q$
Leading terms of the power series for $a_{m}\left(q\right)$ and $b_{m}\left(q\right)$ for $m\leq 6$ are: … The coefficients of the power series of $a_{2n}\left(q\right)$, $b_{2n}\left(q\right)$ and also $a_{2n+1}\left(q\right)$, $b_{2n+1}\left(q\right)$ are the same until the terms in $q^{2n-2}$ and $q^{2n}$, respectively. … Numerical values of the radii of convergence $\rho_{n}^{(j)}$ of the power series (28.6.1)–(28.6.14) for $n=0,1,\dots,9$ are given in Table 28.6.1. Here $j=1$ for $a_{2n}\left(q\right)$, $j=2$ for $b_{2n+2}\left(q\right)$, and $j=3$ for $a_{2n+1}\left(q\right)$ and $b_{2n+1}\left(q\right)$. …
6: 1.3 Determinants
The cofactor $A_{jk}$ of $a_{jk}$ is … For real-valued $a_{jk}$, … where $\omega_{1},\omega_{2},\dots,\omega_{n}$ are the $n$th roots of unity (1.11.21). … If $D_{n}[a_{j,k}]$ tends to a limit $L$ as $n\to\infty$, then we say that the infinite determinant $D_{\infty}[a_{j,k}]$ converges and $D_{\infty}[a_{j,k}]=L$. … Here $\delta_{j,k}$ is the Kronecker delta. …
7: 27.2 Functions
where $p_{1},p_{2},\dots,p_{\nu\left(n\right)}$ are the distinct prime factors of $n$, each exponent $a_{r}$ is positive, and $\nu\left(n\right)$ is the number of distinct primes dividing $n$. … Note that $\sigma_{0}\left(n\right)=d\left(n\right)$. …Note that $J_{1}\left(n\right)=\phi\left(n\right)$. In the following examples, $a_{1},\dots,a_{\nu\left(n\right)}$ are the exponents in the factorization of $n$ in (27.2.1). … Table 27.2.1 lists the first 100 prime numbers $p_{n}$. …
8: 26.12 Plane Partitions
26.12.10 $\left(\prod_{h=1}^{r}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right)\*\left(\prod_% {h=1}^{r+1}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right);$
26.12.11 $\left(\prod_{h=1}^{r+1}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right)\*\left(% \prod_{h=1}^{r}\prod_{j=1}^{s+1}\frac{h+j+t-1}{h+j-1}\right).$
The notation $\sum_{\pi\subseteq B(r,s,t)}$ denotes the sum over all plane partitions contained in $B(r,s,t)$, and $|\pi|$ denotes the number of elements in $\pi$. … where $\sigma_{2}(j)$ is the sum of the squares of the divisors of $j$. …
26.12.26 $\operatorname{pp}\left(n\right)\sim\frac{\left(\zeta\left(3\right)\right)^{7/3% 6}}{2^{11/36}(3\pi)^{1/2}n^{25/36}}\exp\left(3\left(\zeta\left(3\right)\right)% ^{1/3}\left(\tfrac{1}{2}n\right)^{2/3}+\zeta'\left(-1\right)\right),$
9: 24.2 Definitions and Generating Functions
$B_{2n+1}=0$ ,
10: Bibliography
• M. Abramowitz (1949) Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.
• S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.
• A. Apelblat (1989) Derivatives and integrals with respect to the order of the Struve functions $\mathbf{H}_{\nu}(x)$ and $\mathbf{L}_{\nu}(x)$ . J. Math. Anal. Appl. 137 (1), pp. 17–36.
• V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
• R. Askey (1980) Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11 (6), pp. 938–951.