# with respect to summation

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##### 1: 18.3 Definitions
In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_{n}\left(x\right)$, $n=0,1,\dots,N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\dots,N+1$, of $T_{N+1}\left(x\right)$: …
##### 2: 3.11 Approximation Techniques
When $n>0$ and $0\leq j\leq n$, $0\leq k\leq n$, … Now suppose that $X_{k\ell}=0$ when $k\neq\ell$, that is, the functions $\phi_{k}(x)$ are orthogonal with respect to weighted summation on the discrete set $x_{1},x_{2},\dots,x_{J}$. … …
##### 3: 25.11 Hurwitz Zeta Function
The Riemann zeta function is a special case: … For other series expansions similar to (25.11.10) see Coffey (2008). … In (25.11.18)–(25.11.24) primes on $\zeta$ denote derivatives with respect to $s$. … When $a=1$, (25.11.35) reduces to (25.2.3). … uniformly with respect to bounded nonnegative values of $\alpha$. …
##### 4: 1.8 Fourier Series
As $n\to\infty$Then the series (1.8.1) converges to the sum …at every point at which $f(x)$ has both a left-hand derivative (that is, (1.4.4) applies when $h\to 0-$) and a right-hand derivative (that is, (1.4.4) applies when $h\to 0+$). … when $f(x)$ and $g(x)$ are square-integrable and $a_{n},b_{n}$ and $a^{\prime}_{n},b^{\prime}_{n}$ are their respective Fourier coefficients.
##### 5: 23.18 Modular Transformations
according as the elements $\begin{bmatrix}a&b\\ c&d\end{bmatrix}$ of $\mathcal{A}$ in (23.15.3) have the respective forms …Here e and o are generic symbols for even and odd integers, respectively. …
23.18.6 $\varepsilon(\mathcal{A})=\exp\left(\pi i\left(\frac{a+d}{12c}+s(-d,c)\right)% \right),$
23.18.7 ${s(d,c)=\sum_{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\left\lfloor\frac{dr}{c}% \right\rfloor-\frac{1}{2}\right),}$ $c>0$.
##### 6: Errata
• Equations (15.2.3_5), (19.11.6_5)

These equations, originally added in Other Changes and Other Changes, respectively, have been assigned interpolated numbers.

• Subsection 19.25(vi)

The Weierstrass lattice roots $e_{j},$ were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots $e_{j}$, and lattice invariants $g_{2}$, $g_{3}$, now link to their respective definitions (see §§23.2(i), 23.3(i)).

Reported by Felix Ospald.

• Equations (10.15.1), (10.38.1)

These equations have been generalized to include the additional cases of $\ifrac{\partial J_{-\nu}\left(z\right)}{\partial\nu}$, $\ifrac{\partial I_{-\nu}\left(z\right)}{\partial\nu}$, respectively.

• Subsections 14.5(ii), 14.5(vi)

The titles have been changed to $\mu=0$ , $\nu=0,1$ , and Addendum to §14.5(ii)$\mu=0$, $\nu=2$ , respectively, in order to be more descriptive of their contents.

• Equation (8.12.18)
8.12.18 $\rselection{Q\left(a,z\right)\\ P\left(a,z\right)}\sim\frac{z^{a-\frac{1}{2}}{\mathrm{e}}^{-z}}{\Gamma\left(a% \right)}{\left(d(\pm\chi)\sum_{k=0}^{\infty}\frac{A_{k}(\chi)}{z^{k/2}}\mp\sum% _{k=1}^{\infty}\frac{B_{k}(\chi)}{z^{k/2}}\right)}$

The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

Reported 2017-01-28 by Richard Paris.

• ##### 7: 8.12 Uniform Asymptotic Expansions for Large Parameter
in each case uniformly with respect to $\lambda$ in the sector $|\operatorname{ph}\lambda|\leq 2\pi-\delta$ ($<2\pi$). … For the asymptotic behavior of $c_{k}(\eta)$ as $k\to\infty$ see Dunster et al. (1998) and Olde Daalhuis (1998c). … Higher coefficients $A_{k}(\chi)$, $B_{k}(\chi)$, up to $k=8$, are given in Paris (2002b). … For asymptotic expansions, as $a\to\infty$, of the inverse function $x=x(a,q)$ that satisfies the equation …These expansions involve the inverse error function $\operatorname{inverfc}\left(x\right)$7.17), and are uniform with respect to $q\in[0,1]$. …
##### 8: Bibliography G
• F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
• G. Gasper (1981) Orthogonality of certain functions with respect to complex valued weights. Canad. J. Math. 33 (5), pp. 1261–1270.
• W. Gautschi (1984) Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.
• J. J. Gray (2000) Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.
• R. A. Gustafson (1987) Multilateral summation theorems for ordinary and basic hypergeometric series in ${\rm U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.
• ##### 9: 34.7 Basic Properties: $\mathit{9j}$ Symbol
The $\mathit{9j}$ symbol has symmetry properties with respect to permutation of columns, permutation of rows, and transposition of rows and columns; these relate 72 independent $\mathit{9j}$ symbols. …
34.7.4 $\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{m_{r1},m_{r2},r=1,2,3}\begin{pmatrix}j% _{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\*\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}.$
##### 10: Bibliography
• R. W. Abernathy and R. P. Smith (1993) Algorithm 724: Program to calculate F-percentiles. ACM Trans. Math. Software 19 (4), pp. 481–483.
• G. B. Airy (1849) Supplement to a paper “On the intensity of light in the neighbourhood of a caustic”. Trans. Camb. Phil. Soc. 8, pp. 595–599.
• G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.
• 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.
• A. Apelblat (1991) Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42 (5), pp. 708–714.