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11: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

… ►Define the elementary symmetric function $E_s(𝐳)$ by … ►This form of $T_N$ can be applied to (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) if we use …The number of terms in $T_N$ can be greatly reduced by using variables $𝐙=\mathrm{𝟏}-(𝐳/A)$ with $A$ chosen to make $E_1(𝐙)=0$. … ►

12: 18.28 Askey–Wilson Class

… ►The Askey–Wilson polynomials form a system of OP’s $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. The $q$-Racah polynomials form a system of OP’s $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots },N$, that are orthogonal with respect to a weight function on a sequence $\{q^{-y}+c{q}^{y+1}\}$, $y=0,1,\mathrm{\dots },N$, with $c$ a constant. … ►The polynomials $p_n(x;a,b,c,d|q)$ are symmetric in the parameters $a,b,c,d$. … ►Assume $a,b,c,d$ are all real, or two of them are real and two form a conjugate pair, or none of them are real but they form two conjugate pairs. … ►

18.28.29 $$\underset{q\to 1}{lim}\frac{p_n(1-\frac{1}{2}x{(1-q)}^2;q^a,q^b,q^c,q^d|q)}{{(1-q)}^{3n}}=W_n(x;a,b,c,d).$$

ⓘ

Symbols:

$p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial, $W_n(x;a,b,c,d)$: Wilson polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Substitute (18.28.1) in the left-hand side, take the limit, and use (18.26.1).

Notes:

Another form of this limit is given in Koekoek et al. (2010, (14.1.21)).

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E29

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See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

…

13: 19.34 Mutual Inductance of Coaxial Circles

§19.34 Mutual Inductance of Coaxial Circles

… ►

19.34.5 $$\frac{3c^2}{8\pi ab}M=3R_F(0,r_{+}^2,r_{-}^2)-2r_{-}^2{R}_D(0,r_{+}^2,r_{-}^2),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E5

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See also:

Annotations for §19.34 and Ch.19

… ►

19.34.6 $$\frac{c^2}{2\pi }M=(r_{+}^2+r_{-}^2)R_F(0,r_{+}^2,r_{-}^2)-4R_G(0,r_{+}^2,r_{-}^2).$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$

Referenced by:

§19.34

Permalink:

http://dlmf.nist.gov/19.34.E6

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Annotations for §19.34 and Ch.19

►A simpler form of the result is …

14: 35.6 Confluent Hypergeometric Functions of Matrix Argument

… ►

35.6.2 $$\mathrm{\Psi }(a;b;𝐓)=\frac{1}{{\mathrm{\Gamma }}_m\left(a\right)}{\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{\left|𝐈+𝐗\right|}^{b-a-\frac{1}{2}(m+1)}d𝐗,$$ $\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$, $𝐓\in 𝛀$.

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Defines:

$\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind)

Symbols:

$\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.6.E2

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See also:

Annotations for §35.6(i), §35.6 and Ch.35

►

Laguerre Form

… ►

35.6.5 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{b-\frac{1}{2}(m+1)}{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a}{b};𝐒𝐗)d𝐗={\mathrm{\Gamma }}_m\left(b\right){\left|𝐈-𝐒{𝐓}^{-1}\right|}^{-a}{\left|𝐓\right|}^{-b},$$ $𝐓>𝐒$, $\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable, $>$: greater-than for matrices and $m$: positive integer

Referenced by:

§35.6(ii)

Permalink:

http://dlmf.nist.gov/35.6.E5

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Annotations for §35.6(ii), §35.6 and Ch.35

… ►

35.6.7 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a}{b};𝐓)=\mathrm{etr}\left(𝐓\right){}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{b-a}{b};-𝐓).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix and $b$: complex variable

Permalink:

http://dlmf.nist.gov/35.6.E7

Encodings:

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See also:

Annotations for §35.6(ii), §35.6 and Ch.35

►

35.6.8 $${\int }_{𝛀}{\left|𝐓\right|}^{c-\frac{1}{2}(m+1)}\mathrm{\Psi }(a;b;𝐓)d𝐓=\frac{{\mathrm{\Gamma }}_m\left(c\right){\mathrm{\Gamma }}_m\left(a-c\right){\mathrm{\Gamma }}_m\left(c-b+\frac{1}{2}(m+1)\right)}{{\mathrm{\Gamma }}_m\left(a\right){\mathrm{\Gamma }}_m\left(a-b+\frac{1}{2}(m+1)\right)},$$ $\mathrm{\Re }\left(a\right)>\mathrm{\Re }\left(c\right)+\frac{1}{2}(m-1)>m-1$, $\mathrm{\Re }\left(c-b\right)>-1$.

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Symbols:

$\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind), $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.6.E8

Encodings:

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See also:

Annotations for §35.6(ii), §35.6 and Ch.35

…

15: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

Jacobi Form

… ►

Confluent Form

… ►Let $f:𝛀\to ℂ$ (a) be orthogonally invariant, so that $f(𝐓)$ is a symmetric function of $t_1,\mathrm{\dots },t_m$, the eigenvalues of the matrix argument $𝐓\in 𝛀$; (b) be analytic in $t_1,\mathrm{\dots },t_m$ in a neighborhood of $𝐓=𝟎$; (c) satisfy $f(𝟎)=1$. … ►

35.7.10 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha a,\alpha b}{\alpha c};𝐓)$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable and $\alpha \to \mathrm{\infty }$: parameter

Permalink:

http://dlmf.nist.gov/35.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(iv), §35.7 and Ch.35

… ►

35.7.11 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐈-{\alpha }^{-1}𝐓)$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable and $\alpha \to \mathrm{\infty }$: parameter

Permalink:

http://dlmf.nist.gov/35.7.E11

Encodings:

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See also:

Annotations for §35.7(iv), §35.7 and Ch.35

…

16: 1.2 Elementary Algebra

… ►

Special Forms of Square Matrices

… ►a real symmetric matrix if … ►Equation (3.2.7) displays a tridiagonal matrix in index form; (3.2.4) does the same for a lower triangular matrix. … ►The matrix $𝐀$ has a determinant, $det(𝐀)$, explored further in §1.3, denoted, in full index form, as … ►Non-defective matrices are precisely the matrices which can be diagonalized via a similarity transformation of the form …

17: 19.26 Addition Theorems

§19.26 Addition Theorems

… ►Equivalent forms of (19.26.2) are given by …Equivalent forms of (19.26.11) are given by … ►

§19.26(iii) Duplication Formulas

… ►Equivalent forms are given by (19.22.22). …

18: 21.5 Modular Transformations

… ►The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi (𝚪)$ is determinate: …($𝐁$ symmetric with integer elements and even diagonal elements.) …($𝐁$ symmetric with integer elements.) …

19: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►These are based on the Liouville normal form of (1.13.29). … ►

Self-Adjoint and Symmetric Operators

… ►Consider formally self-adjoint operators of the form … ►A linear operator $T$ with dense domain is called symmetric if … ►

Self-adjoint extensions of a symmetric Operator

…

20: 18.39 Applications in the Physical Sciences

… ►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ►The solutions (18.39.8) are called the stationary states as separation of variables in (18.39.9) yields solutions of form … ►defines the potential for a symmetric restoring force $-kx$ for displacements from $x=0$. … ►The orthonormal stationary states and corresponding eigenvalues are then of the form … ►These, taken together with the infinite sets of bound states for each $l$, form complete sets. …