multiplication theorems

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11: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

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17.9.3_5

{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(c/a,c/b;q\right)_{% \infty}}{\left(c,c/(ab);q\right)_{\infty}}{{}_{3}\phi_{2}}\left({a,b,abz/c% \atop qab/c,0};q,q\right)+\frac{\left(a,b,abz/c;q\right)_{\infty}}{\left(c,ab/% c,z;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({c/a,c/b,z\atop qc/(ab),0};q,q% \right),

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Proof sketch:: Derivable from (17.9.13) after replacing $a$ with $d/a$ and then taking $d\rightarrow 0$ . Then replace $a$ , $b$ , $c$ , $e$ with $abz/c$ , $a$ , $b$ , $c$ respectively.
Referenced by:: §17.9(i), Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/17.9.E3_5
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

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17.9.6

{{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,de/(abc)\right)=\frac{\left(e/a,de/(% bc);q\right)_{\infty}}{\left(e,de/(abc);q\right)_{\infty}}{{}_{3}\phi_{2}}% \left({a,d/b,d/c\atop d,de/(bc)};q,e/a\right),

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

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17.9.13

{{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,\frac{de}{abc}\right)=\frac{\left(e/b% ,e/c;q\right)_{\infty}}{\left(e,e/(bc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left% ({d/a,b,c\atop d,bcq/e};q,q\right)+\frac{\left(d/a,b,c,de/(bc);q\right)_{% \infty}}{\left(d,e,bc/e,de/(abc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left({e/b,% e/c,de/(abc)\atop de/(bc),eq/(bc)};q,q\right).

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Referenced by:: (17.9.3_5)
Permalink:: http://dlmf.nist.gov/17.9.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

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17.9.14

{{}_{4}\phi_{3}}\left({q^{-n},a,b,c\atop d,e,f};q,q\right)=\frac{\left(e/a,f/a% ;q\right)_{n}}{\left(e,f;q\right)_{n}}a^{n}{{}_{4}\phi_{3}}\left({q^{-n},a,d/b% ,d/c\atop d,aq^{1-n}/e,aq^{1-n}/f};q,q\right)=\frac{\left(a,ef/(ab),ef/(ac);q% \right)_{n}}{\left(e,f,ef/(abc);q\right)_{n}}{{}_{4}\phi_{3}}\left({q^{-n},e/a% ,f/a,ef/(abc)\atop ef/(ab),ef/(ac),q^{1-n}/a};q,q\right).

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.9.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

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Watson’s $q$ -Analog of Whipple’s Theorem

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12: 10.74 Methods of Computation

… ►To ensure that no zeros are overlooked, standard tools are the phase principle and Rouché’s theorem; see §1.10(iv). … ►

Multiple Zeros

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13: 5.5 Functional Relations

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§5.5(iii) Multiplication

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Gauss’s Multiplication Formula

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5.5.7

\prod_{k=1}^{n-1}\Gamma\left(\frac{k}{n}\right)=(2\pi)^{(n-1)/2}n^{-1/2}.

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

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§5.5(iv) Bohr–Mollerup Theorem

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14: 1.9 Calculus of a Complex Variable

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DeMoivre’s Theorem

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Jordan Curve Theorem

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Cauchy’s Theorem

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Liouville’s Theorem

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Dominated Convergence Theorem

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15: 10.18 Modulus and Phase Functions

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10.18.17

{M_{\nu}}^{2}\left(x\right)\sim\frac{2}{\pi x}\left(1+\frac{1}{2}\frac{\mu-1}{% (2x)^{2}}+\frac{1\cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2x)^{4}}+\frac{1% \cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{(\mu-1)(\mu-9)(\mu-25)}{(2x)^{6}}+\dotsb% \right),

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Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
Proof sketch:: See Nemes (2021, Theorem 2.1)
A&S Ref:: 9.2.28
Referenced by:: §10.18(iii), §10.18(iii), §10.40(i), §10.40(i), §10.49(iv), Erratum (V1.1.8) for Section 10.18(iii)
Permalink:: http://dlmf.nist.gov/10.18.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §10.18(iii), §10.18 and Ch.10

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10.18.18

\theta_{\nu}\left(x\right)\sim x-\left(\frac{1}{2}\nu+\frac{1}{4}\right)\pi+% \frac{\mu-1}{2(4x)}+\frac{(\mu-1)(\mu-25)}{6(4x)^{3}}+\frac{(\mu-1)(\mu^{2}-11% 4\mu+1073)}{5(4x)^{5}}+\frac{(\mu-1)(5\mu^{3}-1535\mu^{2}+54703\mu-3\;75733)}{% 14(4x)^{7}}+\dotsb.

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Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
Proof sketch:: See Nemes (2021, Theorem 2.1)
A&S Ref:: 9.2.29
Referenced by:: §10.18(iii), §10.18(iii), Erratum (V1.1.8) for Section 10.18(iii)
Permalink:: http://dlmf.nist.gov/10.18.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §10.18(iii), §10.18 and Ch.10

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16: Bibliography

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W. A. Al-Salam (1990) Characterization theorems for orthogonal polynomials. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 1–24.

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External Links:: Link, MathReview (Walter Van Assche), ZentralBlatt
Referenced by:: §18.3.
Encodings:: BibTeX

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H. Alzer and S. Qiu (2004) Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172 (2), pp. 289–312.

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External Links:: ISSN 0377-0427, Document, MathReview (Živorad Tomovski), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

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G. E. Andrews (1984) Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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External Links:: ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §17.12.
Encodings:: BibTeX

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T. M. Apostol (1952) Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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External Links:: MathReview (N. G. de Bruijn), ZentralBlatt
Referenced by:: (25.11.41), (25.11.42), §25.11(xii).
Encodings:: BibTeX

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T. M. Apostol (2000) A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.

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External Links:: MathReview (Giovanni Coppola), ZentralBlatt
Referenced by:: (25.16.2), (25.16.4), §25.16(i), §25.16.
Encodings:: BibTeX

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17: 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. … ►

§18.36(iii) Multiple Orthogonal Polynomials

… ►Orthogonality of the the classical OP’s with respect to a positive weight function, as in Table 18.3.1 requires, via Favard’s theorem,

A_{n}A_{n-1}C_{n}>0

for

n\geq 1

as per (18.2.9_5). … ►In §18.39(i) it is seen that the functions,

\sqrt{w(x)}\hat{H}_{n+3}\left(x\right)

, are solutions of a Schrödinger equation with a rational potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …

18: Bibliography D

… ►

H. Davenport (2000) Multiplicative Number Theory. 3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.

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Notes:: Revised and with a preface by Hugh L. Montgomery
External Links:: ISBN 0-387-95097-4, MathReview, ZentralBlatt
Referenced by:: §27.12.
Encodings:: BibTeX

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S. C. Dhar (1940) Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.

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External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

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K. Dilcher (2008) On multiple zeros of Bernoulli polynomials. Acta Arith. 134 (2), pp. 149–155.

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External Links:: ISSN 0065-1036, Document, MathReview (Arnold M. Adelberg), ZentralBlatt
Referenced by:: §24.12(iv).
Encodings:: BibTeX

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H. Ding, K. I. Gross, and D. St. P. Richards (1996) Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.

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External Links:: ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt
Referenced by:: §35.7(ii), §35.8(iv), §35.8(v).
Encodings:: BibTeX

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J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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External Links:: ZentralBlatt
Referenced by:: §15.4(ii).
Encodings:: BibTeX

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19: 27.8 Dirichlet Characters

… ►If

k

(>1)

is a given integer, then a function

\chi\left(n\right)

is called a Dirichlet character (mod

k

) if it is completely multiplicative, periodic with period

k

, and vanishes when

\left(n,k\right)>1

. … ►For any character

\chi\pmod{k}

\chi\left(n\right)\neq 0

if and only if

\left(n,k\right)=1

, in which case the Euler–Fermat theorem (27.2.8) implies

\left(\chi\left(n\right)\right)^{\phi\left(k\right)}=1

. …

20: 22.8 Addition Theorems

§22.8 Addition Theorems

… ►If sums/differences of the

z_{j}

’s are rational multiples of

K\left(k\right)

, then further relations follow. …