multiplication%20theorems

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1: 27.2 Functions

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§27.2(i) Definitions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. … ►If $\left(a,n\right)=1$, then the Euler–Fermat theorem states that …

2: Bibliography K

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On multiple zeros of derivatives of Bessel’s cylindrical functions. Dokl. Akad. Nauk SSSR 288 (2), pp. 285–288 (Russian).

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Notes:

In Russian. English translation: Soviet Math. Dokl. 33(3), 650–653, (1986).

External Links:

ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(i), §10.74(vi).

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On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 27(6), pp. 18–25

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ISSN 0044-4669, Document, MathReview (Hans-Jürgen Albrand), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi), 5th item.

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Multiple complex zeros of derivatives of the cylindrical Bessel functions. Dokl. Akad. Nauk SSSR 299 (3), pp. 614–618 (Russian).

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Notes:

In Russian. English translation: Soviet Phys. Dokl. 33(3),196–198, (1988).

External Links:

ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi).

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A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§30.10.

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A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

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External Links:

Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(i).

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3: Bibliography M

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Asymptotic expansion of a multiple integral. SIAM J. Math. Anal. 18 (6), pp. 1630–1637.

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ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§2.5(ii).

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A $q$-analog of the ${}_5{}^{}F_4^{}(1)$ summation theorem for hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ . Adv. in Math. 57 (1), pp. 14–33.

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ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§17.15.

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

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On the evaluation of some multiple series. J. London Math. Soc. (2) 33, pp. 368–371.

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External Links:

Document, MathReview (V. F. Cowling), ZentralBlatt

Referenced by:

(25.6.5).

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4: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ►has $n$ zeros in $ℂ$, counting each zero according to its multiplicity. … ► … ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

5: Bibliography D

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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.

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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).

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

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ISSN 0065-1036, Document, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.12(iv).

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Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.

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ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt

Referenced by:

§35.7(ii), §35.8(iv), §35.8(v).

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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).

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