constant term identities

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1: 17.14 Constant Term Identities

§17.14 Constant Term Identities

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Rogers–Ramanujan Constant Term Identities

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2: 18.2 General Orthogonal Polynomials

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§18.2(iii) Standardization and Related Constants

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Constants

… ►(i) the traditional OP standardizations of Table 18.3.1, where each is defined in terms of the above constants. … ►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). … ►As in §18.1(i) we assume that $p_{-1}(x)\equiv 0$. …

3: 18.39 Applications in the Physical Sciences

… ►Here the term $-\frac{{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}$ represents the quantum kinetic energy of a single particle of mass $m$, and $V(x)$ its potential energy. … ►The spectrum is mixed as in §1.18(viii), with the discrete eigenvalues given by (18.39.18) and the continuous eigenvalues by ${(\alpha \mathrm{\hslash }\gamma )}^2/(2m)$ ($\gamma \ge 0$) with corresponding eigenfunctions ${\mathrm{e}}^{\alpha (x-x_e)/2}{W}_{\lambda ,\mathrm{i}\gamma }\left(2\lambda {\mathrm{e}}^{-\alpha (x-x_e)}\right)$ expressed in terms of Whittaker functions (13.14.3). … ►

a) Spherical Radial Coulomb Wave Functions Expressed in terms of Laguerre OP’s

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d) Radial Coulomb Wave Functions Expressed in Terms of the Associated Coulomb–Laguerre OP’s

… ►which corresponds to the exact results, in terms of Whittaker functions, of §§33.2 and 33.14, in the sense that projections onto the functions ${\varphi }_{n,l}(sr)/r$, the functions bi-orthogonal to ${\varphi }_{n,l}(sr)$, are identical. …

4: 18.36 Miscellaneous Polynomials

… ►These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. … ►The possibility of generalization to $\alpha =-k$, for $k\in \mathbb{N}$, is implicit in the identity Szegő (1975, page 102), … ►

18.36.7 $$T_k(y)\equiv -x{y}^{\prime \prime }+\frac{x-k}{x+k}((x+k+1)y^{\prime }-y)=(n-1)y.$$

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Symbols:

$\equiv$: equals by definition, $y$: real variable, $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.36(vi)

Permalink:

http://dlmf.nist.gov/18.36.E7

Encodings:

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

►The restriction to $n\ge 1$ is now apparent: (18.36.7) does not posses a solution if $y(x)$ is a constant. … ►Hermite EOP’s are defined in terms of classical Hermite OP’s. …

5: 16.4 Argument Unity

… ►See Erdélyi et al. (1953a, §4.4(4)) for a non-terminating balanced identity. … ►

§16.4(iii) Identities

… ►There are two types of three-term identities for ${}_3{}^{}F_2^{}$’s. …Methods of deriving such identities are given by Bailey (1964), Rainville (1960), Raynal (1979), and Wilson (1978). … ► …

6: 2.10 Sums and Sequences

… ►Formula (2.10.2) is useful for evaluating the constant term in expansions obtained from (2.10.1). … ►The formula for summation by parts is … ►For extensions to $\alpha \le 0$, higher terms, and other examples, see Olver (1997b, Chapter 8). … ►From the identities … ►For higher terms see §18.15(iii). …

7: 2.4 Contour Integrals

… ►Then … … ►Then the Laplace transform … ►If $p^{\prime }(t_0)\ne 0$, then $\mu =1$, $\lambda$ is a positive integer, and the two resulting asymptotic expansions are identical. Thus the right-hand side of (2.4.14) reduces to the error terms. …

8: 2.11 Remainder Terms; Stokes Phenomenon

§2.11 Remainder Terms; Stokes Phenomenon

… ►The error term is, in fact, approximately 700 times the last term obtained in (2.11.4). … ►From (2.11.5) and the identity … ►The numerically smallest terms are the 5th and 6th. Truncation after 5 terms yields 0. …

9: 25.5 Integral Representations

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§25.5(i) In Terms of Elementary Functions

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25.5.6 $$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{{\mathrm{e}}^x}dx,$$ $\mathrm{\Re }s>-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Proof sketch:

Derivable from (25.5.1) by first replacing $x^{s-1}\mapsto x^{s-1}(1+{\mathrm{e}}^{-x}-{\mathrm{e}}^{-x})$, using the identity ${\mathrm{e}}^{-x}=(1-{\mathrm{e}}^{-x})/({\mathrm{e}}^x-1)$, and replacing ${({\mathrm{e}}^x-1)}^{-1}\mapsto ({({\mathrm{e}}^x-1)}^{-1}-1/x+1/2-1/2+1/x)$ in the resulting integrand with (5.2.1), (5.5.1).

Referenced by:

(25.5.7)

Permalink:

http://dlmf.nist.gov/25.5.E6

Encodings:

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See also:

Annotations for §25.5(i), §25.5 and Ch.25

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§25.5(ii) In Terms of Other Functions

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25.5.14 $$\omega (x)\equiv \sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2\pi x}=\frac{1}{2}\left({\theta }_3\left(0\right|\mathrm{i}x)-1\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $x$: real variable

Keywords:

definition

Source:

Titchmarsh (1986b, p. 22)

Permalink:

http://dlmf.nist.gov/25.5.E14

Encodings:

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See also:

Annotations for §25.5(ii), §25.5 and Ch.25

… ►In (25.5.15)–(25.5.19), , $\psi \left(x\right)$ is the digamma function, and $\gamma$ is Euler’s constant (§5.2). …

10: 23.20 Mathematical Applications

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23.20.1 $$C:y^2=x^3+ax+b,$$

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Symbols:

$x$: real part of $z$ , $y$: imaginary part of $z$ , $C$: elliptic curve, $a$: constant and $b$: constant

Permalink:

http://dlmf.nist.gov/23.20.E1

Encodings:

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See also:

Annotations for §23.20(ii), §23.20 and Ch.23

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23.20.2 $$C:y^{2}z=x^3+ax{z}^2+b{z}^3,$$

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Symbols:

$x$: real part of $z$ , $y$: imaginary part of $z$ , $z$: complex, $C$: elliptic curve, $a$: constant and $b$: constant

Permalink:

http://dlmf.nist.gov/23.20.E2

Encodings:

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See also:

Annotations for §23.20(ii), §23.20 and Ch.23

… ►Here $a$ and $b$ are real or complex constants. … ►In terms of $(x,y)$ the addition law can be expressed $(x,y)+o=(x,y)$, $(x,y)+(x,-y)=o$; otherwise $(x_1,y_1)+(x_2,y_2)=(x_3,y_3)$, where … ►If $a,b\in ℝ$, then $C$ intersects the plane ${ℝ}^2$ in a curve that is connected if $\mathrm{\Delta }\equiv 4a^3+27b^2>0$; if , then the intersection has two components, one of which is a closed loop. …