indices differing by an integer

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1: 2.7 Differential Equations

… ►when $s=1,2,3,\mathrm{\dots }$. …

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 $\ast$ 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 $\ast$ 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

… ► ►►►►►►►

$𝕃$	lattice in $ℂ$.
$\mathrm{\ell },n$	integers.
$m$	integer, except in §23.20(ii).
…
primes	derivatives with respect to the variable, except where indicated otherwise.
…
$n\mathbb{Z}$	set of all integer multiples of $n$.
$S_1/S_2$	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).)
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4: 21.1 Special Notation

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$g,h$	positive integers.
${\mathbb{Z}}^g$	$\mathbb{Z}\times \mathbb{Z}\times \mathrm{\cdots }\times \mathbb{Z}$ ($g$ times).
…
${\mathbb{Z}}^{g\times h}$	set of all $g\times h$ matrices with integer elements.
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$S_1/S_2$	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).)
$a\circ b$	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 $-1$.
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5: 20.1 Special Notation

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$m$, $n$	integers.
$z$ $(\in ℂ)$	the argument.
…
$S_1/S_2$	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 $0\le m\le q$ and $0\le n\le p$, and none of $a_k-b_j$ is a positive integer when $1\le k\le n$ and $1\le j\le m$. … ►

16.17.1 $$G_{p,q}^{m,n}(z;𝐚;𝐛)=G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})=\frac{1}{2\pi \mathrm{i}}{\int }_L\left(\prod _{\mathrm{\ell }=1}^m\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)\prod _{\mathrm{\ell }=1}^n\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)/\left(\prod _{\mathrm{\ell }=m}^{q-1}\mathrm{\Gamma }\left(1-b_{\mathrm{\ell }+1}+s\right)\prod _{\mathrm{\ell }=n}^{p-1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }+1}-s\right)\right)\right)z^{s}ds,$$

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Defines:

$G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

Figure 16.17.1, Figure 16.17.1, §16.19

Permalink:

http://dlmf.nist.gov/16.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.17 and Ch.16

… ►Assume $p\le q$, no two of the bottom parameters $b_j$, $j=1,\mathrm{\dots },m$, differ by an integer, and $a_j-b_k$ is not a positive integer when $j=1,2,\mathrm{\dots },n$ and $k=1,2,\mathrm{\dots },m$. …where $\ast$ indicates that the entry $1+b_k-b_k$ is omitted. … ►

16.17.3 $$A_{p,q,k}^{m,n}(z)=\prod _{\begin{array}{c}\mathrm{\ell }=1\\ \mathrm{\ell }\ne k\end{array}}^m\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-b_k\right)\prod _{\mathrm{\ell }=1}^n\mathrm{\Gamma }\left(1+b_k-a_{\mathrm{\ell }}\right)z^{b_k}/\left(\prod _{\mathrm{\ell }=m}^{q-1}\mathrm{\Gamma }\left(1+b_k-b_{\mathrm{\ell }+1}\right)\prod _{\mathrm{\ell }=n}^{p-1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }+1}-b_k\right)\right).$$

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Defines:

$A_{p,q}^{m,n}(z)$: coefficient (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.17 and Ch.16

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 $𝐅$ is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as $𝐅$ is an entire function of each of its parameters $a$, $b$, and $c$:​ this results in fewer restrictions and simpler equations. … ► ►►►

$ℂ$	complex plane (excluding infinity).
…
$\mathrm{\Delta }$ (or ${\mathrm{\Delta }}_x$)	forward difference operator: $\mathrm{\Delta }f(x)=f(x+1)-f(x)$.
…

… ►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 $F(a,b;c;z)$ and $𝐅(a,b;c;z)$ 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 $𝐅(a,b;c;z)$ 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 $\{0,1,\mathrm{\dots },N\}$ or $\{0,1,2,\mathrm{\dots }\}$, then the role of the differentiation operator $d/dx$ in the case of classical OP’s (§18.3) is played by ${\mathrm{\Delta }}_x$, 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). …