# indices differing by an integer

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## 1—10 of 16 matching pages

##### 1: 2.7 Differential Equations

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►when $s=1,2,3,\mathrm{\dots}$.
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##### 2: 16.8 Differential Equations

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►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.
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►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.
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►When $p=q+1$ and some of the ${a}_{j}$
differ by an integer a limiting process can again be applied.
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##### 3: 23.1 Special Notation

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$\mathbb{L}$ | lattice in $\u2102$. |
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$\mathrm{\ell},n$ | integers. |

$m$ | integer, except in §23.20(ii). |

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primes | derivatives with respect to the variable, except where indicated otherwise. |

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$n\mathbb{Z}$ | 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. |
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${\mathbb{Z}}^{g}$ | $\mathbb{Z}\times \mathbb{Z}\times \mathrm{\cdots}\times \mathbb{Z}$ ($g$ times). |

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${\mathbb{Z}}^{g\times h}$ | 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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►Primes on the $\theta $ symbols indicate derivatives with respect to the argument of the $\theta $ function.
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$m$, $n$ | integers. |
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$z$ $(\in \u2102)$ | the argument. |

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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).) |

##### 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

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►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 $1\le j\le m$.
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16.17.1
$${G}_{p,q}^{m,n}(z;\mathbf{a};\mathbf{b})={G}_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{{a}_{1},\mathrm{\dots},{a}_{p}}{{b}_{1},\mathrm{\dots},{b}_{q}})=\frac{1}{2\pi \mathrm{i}}{\int}_{L}\left(\prod _{\mathrm{\ell}=1}^{m}\mathrm{\Gamma}\left({b}_{\mathrm{\ell}}-s\right)\prod _{\mathrm{\ell}=1}^{n}\mathrm{\Gamma}\left(1-{a}_{\mathrm{\ell}}+s\right)/\left(\prod _{\mathrm{\ell}=m}^{q-1}\mathrm{\Gamma}\left(1-{b}_{\mathrm{\ell}+1}+s\right)\prod _{\mathrm{\ell}=n}^{p-1}\mathrm{\Gamma}\left({a}_{\mathrm{\ell}+1}-s\right)\right)\right){z}^{s}ds,$$

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

##### 8: Mathematical Introduction

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►Two other ways in which this Handbook differs from AMS 55, and other handbooks, are as follows.
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►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).
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►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.
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►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.
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$\u2102$ | complex plane (excluding infinity). |
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$\mathrm{\Delta}$ (or ${\mathrm{\Delta}}_{x}$) | forward difference operator: $\mathrm{\Delta}f(x)=f(x+1)-f(x)$. |

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##### 9: 15.2 Definitions and Analytical Properties

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*Except where indicated otherwise*principal branches of $F(a,b;c;z)$ and $\mathbf{F}(a,b;c;z)$ 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 $\mathbf{F}(a,b;c;z)$ 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

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►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). …