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1: 2.7 Differential Equations
when s = 1 , 2 , 3 , . …
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
𝕃 lattice in .
, n integers.
m integer, except in §23.20(ii).
primes derivatives with respect to the variable, except where indicated otherwise.
n 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).)
4: 21.1 Special Notation
g , h positive integers.
g × × × ( g times).
g × h set of all g × h matrices with integer elements.
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 b intersection index of a and b , two cycles lying on a closed surface. a b = 0 if a and b do not intersect. Otherwise a 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.
z ( ) 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 θ symbols indicate derivatives with respect to the argument of the θ 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 m q and 0 n p , and none of a k - b j is a positive integer when 1 k n and 1 j m . …
16.17.1 G p , q m , n ( z ; a ; b ) = G p , q m , n ( z ; a 1 , , a p b 1 , , b q ) = 1 2 π i L ( = 1 m Γ ( b - s ) = 1 n Γ ( 1 - a + s ) / ( = m q - 1 Γ ( 1 - b + 1 + s ) = n p - 1 Γ ( a + 1 - s ) ) ) z s d s ,
Assume p q , no two of the bottom parameters b j , j = 1 , , m , differ by an integer, and a j - b k is not a positive integer when j = 1 , 2 , , n and k = 1 , 2 , , m . …where * indicates that the entry 1 + b k - b k is omitted. …
16.17.3 A p , q , k m , n ( z ) = = 1 k m Γ ( b - b k ) = 1 n Γ ( 1 + b k - a ) z b k / ( = m q - 1 Γ ( 1 + b k - b + 1 ) = n p - 1 Γ ( a + 1 - b k ) ) .
8: 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 , , N } or { 0 , 1 , 2 , } , then the role of the differentiation operator d / d x in the case of classical OP’s (§18.3) is played by Δ x , the forward-difference operator, or by x , the backward-difference operator; compare §18.1(i). … If the orthogonality interval is ( - , ) or ( 0 , ) , then the role of d / d x can be played by δ x , the central-difference operator in the imaginary direction (§18.1(i)). … All n zeros of an OP p n ( x ) are simple, and they are located in the interval of orthogonality ( a , b ) . …
9: 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. …
complex plane (excluding infinity).
Δ (or Δ x ) forward difference operator: Δ 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. …
10: 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 . …