# indices differing by an integer

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##### 1: 2.7 Differential Equations
when $s=1,2,3,\dots$. …
##### 2: 16.8 Differential Equations
When no $b_{j}$ is an integer, and no two $b_{j}$ differ by an integer, a fundamental set of solutions of (16.8.3) is given by …where $*$ indicates that the entry $1+b_{j}-b_{j}$ is omitted. … When $p=q+1$, and no two $a_{j}$ differ by an integer, another fundamental set of solutions of (16.8.3) is given by …where $*$ indicates that the entry $1-a_{j}+a_{j}$ is omitted. … When $p=q+1$ and some of the $a_{j}$ differ by an integer a limiting process can again be applied. …
##### 3: 23.1 Special Notation
 $\mathbb{L}$ lattice in $\mathbb{C}$. integers. integer, except in §23.20(ii). … derivatives with respect to the variable, except where indicated otherwise. … set of all integer multiples of $n$. set of all elements of $S_{1}$, modulo elements of $S_{2}$. Thus two elements of $S_{1}/S_{2}$ are equivalent if they are both in $S_{1}$ and their difference is in $S_{2}$. (For an example see §20.12(ii).) …
##### 4: 21.1 Special Notation
 $g,h$ positive integers. $\mathbb{Z}\times\mathbb{Z}\times\cdots\times\mathbb{Z}$ ($g$ times). … set of all $g\times h$ matrices with integer elements. … set of all elements of $S_{1}$, modulo elements of $S_{2}$. Thus two elements of $S_{1}/S_{2}$ are equivalent if they are both in $S_{1}$ and their difference is in $S_{2}$. (For an example see §20.12(ii).) intersection index of $a$ and $b$, two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$. …
##### 5: 20.1 Special Notation
 $m$, $n$ integers. the argument. … set of all elements of $S_{1}$, modulo elements of $S_{2}$. Thus two elements of $S_{1}/S_{2}$ are equivalent if they are both in $S_{1}$ and their difference is in $S_{2}$. (For an example see §20.12(ii).)
Primes on the $\theta$ symbols indicate derivatives with respect to the argument of the $\theta$ function. …
##### 6: 2.9 Difference Equations
###### §2.9 Difference Equations
or equivalently the second-order homogeneous linear difference equation … For analogous results for difference equations of the form … For an introduction to, and references for, the general asymptotic theory of linear difference equations of arbitrary order, see Wimp (1984, Appendix B). …
##### 7: 16.17 Definition
Assume also that $m$ and $n$ are integers such that $0\leq m\leq q$ and $0\leq n\leq p$, and none of $a_{k}-b_{j}$ is a positive integer when $1\leq k\leq n$ and $1\leq j\leq m$. …
16.17.1 ${G^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right)={G^{m,n}_{p,q}}\left(z;{a_% {1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\frac{1}{2\pi\mathrm{i}}\int_{L% }\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{m}\Gamma\left(b_{\ell}-s\right% )\prod\limits_{\ell=1}^{n}\Gamma\left(1-a_{\ell}+s\right)}{\left(\prod\limits_% {\ell=m}^{q-1}\Gamma\left(1-b_{\ell+1}+s\right)\prod\limits_{\ell=n}^{p-1}% \Gamma\left(a_{\ell+1}-s\right)\right)}}\right)z^{s}\,\mathrm{d}s,$
Assume $p\leq q$, no two of the bottom parameters $b_{j}$, $j=1,\dots,m$, differ by an integer, and $a_{j}-b_{k}$ is not a positive integer when $j=1,2,\dots,n$ and $k=1,2,\dots,m$. …where $*$ indicates that the entry $1+b_{k}-b_{k}$ is omitted. …
16.17.3 $A_{p,q,k}^{m,n}(z)=\ifrac{\prod\limits_{\begin{subarray}{c}\ell=1\\ \ell\neq k\end{subarray}}^{m}\Gamma\left(b_{\ell}-b_{k}\right)\prod\limits_{% \ell=1}^{n}\Gamma\left(1+b_{k}-a_{\ell}\right)z^{b_{k}}}{\left(\prod\limits_{% \ell=m}^{q-1}\Gamma\left(1+b_{k}-b_{\ell+1}\right)\prod\limits_{\ell=n}^{p-1}% \Gamma\left(a_{\ell+1}-b_{k}\right)\right)}.$
##### 8: Mathematical Introduction
Two other ways in which this Handbook differs from AMS 55, and other handbooks, are as follows. … These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … This is because $\mathbf{F}$ is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as $\mathbf{F}$ is an entire function of each of its parameters $a$, $b$, and $c$:​ this results in fewer restrictions and simpler equations. …
 $\mathbb{C}$ complex plane (excluding infinity). … forward difference operator: $\Delta f(x)=f(x+1)-f(x)$. …
However, in many cases the coloring of the surface is chosen instead to indicate the quadrant of the plane to which the phase of the function belongs, thereby achieving a 4D effect. …
##### 9: 15.2 Definitions and Analytical Properties
Except where indicated otherwise principal branches of $F\left(a,b;c;z\right)$ and $\mathbf{F}\left(a,b;c;z\right)$ are assumed throughout the DLMF. The difference between the principal branches on the two sides of the branch cut (§4.2(i)) is given by … The principal branch of $\mathbf{F}\left(a,b;c;z\right)$ is an entire function of $a$, $b$, and $c$. … Let $m$ be a nonnegative integer. …The right-hand side can be seen as an analytical continuation for the left-hand side when $a$ approaches $-m$. …
##### 10: 18.2 General Orthogonal Polynomials
It is assumed throughout this chapter that for each polynomial $p_{n}(x)$ that is orthogonal on an open interval $(a,b)$ the variable $x$ is confined to the closure of $(a,b)$ unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) …
###### §18.2(ii) $x$-Difference Operators
If the orthogonality discrete set $X$ is $\{0,1,\dots,N\}$ or $\{0,1,2,\dots\}$, then the role of the differentiation operator $\ifrac{\mathrm{d}}{\mathrm{d}x}$ in the case of classical OP’s (§18.3) is played by $\Delta_{x}$, the forward-difference operator, or by $\nabla_{x}$, the backward-difference operator; compare §18.1(i). … The constant function $p_{0}(x)$ will often, but not always, be identically $1$ (see, for example, (18.2.11_8)), $p_{-1}(x)=0$ in all cases, by convention, as indicated in §18.1(i). … where the first indicates that the indices of the recursion coefficients $\alpha_{n}$, $\beta_{n}$ of (18.2.31) have been incremented by $1$, when compared to those of (18.2.11_5). …