# multiples of argument

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## 1—10 of 16 matching pages

##### 3: 17.4 Basic Hypergeometric Functions
17.4.6 $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)=\sum_{m,n\geq 0}\frac% {\left(a;q\right)_{m+n}\left(b;q\right)_{m}\left(b^{\prime};q\right)_{n}x^{m}y% ^{n}}{\left(q,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n}},$
17.4.7 $\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)=\sum_{m,n\geq 0}\frac% {\left(a,b;q\right)_{m}\left(a^{\prime},b^{\prime};q\right)_{n}x^{m}y^{n}}{% \left(q;q\right)_{m}\left(q;q\right)_{n}\left(c;q\right)_{m+n}},$
17.4.8 $\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)=\sum_{m,n\geq 0}\frac{\left(a,b;% q\right)_{m+n}x^{m}y^{n}}{\left(q,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n% }}.$
##### 4: 35.10 Methods of Computation
###### §35.10 Methods of Computation
Other methods include numerical quadrature applied to double and multiple integral representations. See Yan (1992) for the ${{}_{1}F_{1}}$ and ${{}_{2}F_{1}}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $\mathbf{O}(m)$ applied to a generalization of the integral (35.5.8). …
##### 5: 17.11 Transformations of $q$-Appell Functions
17.11.1 $\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)=\frac{\left(a,bx,b^{\prime}y;q% \right)_{\infty}}{\left(c,x,y;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({c/a,x,y% \atop bx,b^{\prime}y};q,a\right),$
17.11.2 $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)=\frac{\left(b,ax;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a,b% ^{\prime};q\right)_{n}\left(c/b,x;q\right)_{r}b^{r}y^{n}}{\left(q,c^{\prime};q% \right)_{n}\left(q;q\right)_{r}\left(ax;q\right)_{n+r}},$
17.11.3 $\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)=\frac{\left(a,bx;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a^{% \prime},b^{\prime};q\right)_{n}\left(x;q\right)_{r}\left(c/a;q\right)_{n+r}a^{% r}y^{n}}{\left(q,c/a;q\right)_{n}\left(q,bx;q\right)_{r}}.$
##### 7: Errata
• Equation (18.27.6)

18.27.6 $P^{(\alpha,\beta)}_{n}\left(x;c,d;q\right)=\frac{c^{n}q^{-(\alpha+1)n}\left(q^% {\alpha+1},-q^{\alpha+1}c^{-1}d;q\right)_{n}}{\left(q,-q;q\right)_{n}}\*P_{n}% \left(q^{\alpha+1}c^{-1}x;q^{\alpha},q^{\beta},-q^{\alpha}c^{-1}d;q\right)$

Originally the first argument to 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.

• ##### 8: 10.18 Modulus and Phase Functions
###### §10.18(iii) Asymptotic Expansions for Large Argument
The remainder after $k$ terms in (10.18.17) does not exceed the $(k+1)$th term in absolute value and is of the same sign, provided that $k>\nu-\tfrac{1}{2}$.
##### 9: 18.27 $q$-Hahn Class
18.27.6 $P^{(\alpha,\beta)}_{n}\left(x;c,d;q\right)=\frac{c^{n}q^{-(\alpha+1)n}\left(q^% {\alpha+1},-q^{\alpha+1}c^{-1}d;q\right)_{n}}{\left(q,-q;q\right)_{n}}\*P_{n}% \left(q^{\alpha+1}c^{-1}x;q^{\alpha},q^{\beta},-q^{\alpha}c^{-1}d;q\right).$
##### 10: 1.10 Functions of a Complex Variable
An analytic function $f(z)$ has a zero of order (or multiplicity) $m$ ($\geq\!1$) at $z_{0}$ if the first nonzero coefficient in its Taylor series at $z_{0}$ is that of $(z-z_{0})^{m}$. … … If $-n$ is the first negative integer (counting from $-\infty$) with $a_{-n}\not=0$, then $z_{0}$ is a pole of order (or multiplicity) $n$. …
###### Phase (or Argument) Principle
each location again being counted with multiplicity equal to that of the corresponding zero or pole. …