A generalized Binomial theorem and a summation formulae

V. N. Chubarikov

Čebyševskij sbornik, Tome 21 (2020) no. 4, pp. 270-301.

Résumé

The paper is based on the Binomial theorem and its generalizations to the polynomials of binomial type. Thus, we give some applications to the generalized Waring problemm (Loo-Keng Hua) and Hilbert-Kamke problem (G.I.Arkhipov). We also prove Taylor-Maclaurin formula for the polynomials and smooth functions and give its applications to the numerical analysis (Newton's root-finding algorithm, Hensel lemma in full non-archimedian fields, approximate evaluaion of the function at given point). Next, we prove an analogue of Binomial theorem for Bernoulli polynomials, Euler-Maclaurin summation formula over integers and Poisson summation formula for the lattice and consider some examples of binomial-type polynomials (monomials, rising and falling factorials, Abel and Laguerre polynomials). We prove some binomial properties op Appel and Euler polynomials and establish the multidimensional Taylor formula and the analogues of Euler-Maclaurin and Poisson summation formulas over the lattices. Finally, we consider the multidimensional analogues of these formulas for the multidimensional complex space and prove some properties of binomial-type polynomials of several variables.

Détail

Keywords: the Newton binomial formula, a sequence of the binomial type polynomials, lower and upper factoriales, the Abel, Laguerre, Appell, Bernoulli, Euler polynomials, the Taylor–Maclauren formula, the Euler–Maclauren formula.

Bibliographie

[1] Pincherle S., Amaldi, Le operazioni distributive e le loro applicazioni all'analisi (in collborazione U. Amaldi), Zanichelli, Bologna, 1900

[2] Hurwitz A., “Über Abel's Verallgemeinerung der binomischen Formel”, Acta Math., 26 (1902), 199–203 | DOI | MR

[3] Sheffer I. M., “Some properties of polynomials of type zero”, Duke Math., 5 (1939), 590–622 | DOI | MR | Zbl

[4] Stefensen J. F., “The Poweroid, an Extension of the Mathematical of Power”, Acta Math., 73 (1941), 333–366 | DOI | MR

[5] Touchard J., “Nombres Exponentiels et Nombres de Bernuolli”, Canad. J. Math., 8 (1956), 305–320 | DOI | MR | Zbl

[6] Riordan Dzh., Vvedenie v kombinatornyi analiz, Izd-vo in. lit., M., 1963

[7] Riordan J., “Inverse Relations and Combinatorial Identities”, Amer. Math. Monthly, 71 (1964), 485–498 | DOI | MR | Zbl

[8] Riordan J., Combinatorial Identities, Wiley, New York, 1968 | MR | Zbl

[9] Mullin R., Rota G.-C., “On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration”, Graph Theory and its Applications, 1970, 168–213 | MR

[10] Vinogradov I. M., Metod trigonometricheskikh summ v teorii chisel, 2-e izd., ispravlennoe i dopolnennoe, Nauka, M., 1980, 144 pp.

[11] Karatsuba A. A., Osnovy analiticheskoi teorii chisel, 2-e izd., Nauka, M., 1983, 240 pp. | MR

[12] Hua Loo-Keng, Selected Papers, N.Y.–Heidelberg–Berlin, 1983, 888 pp. | MR | Zbl

[13] Arkhipov G. I., Izbrannye trudy, Izd-vo Orlovskogo un-ta, Orel, 2013, 464 pp.

[14] Arkhipov G. I., Sadovnichii V. A., Chubarikov V. N., Lektsii po matematicheskomu analizu, 4-e izd., ispr., Drofa, M., 2004, 640 pp.

[15] Arkhipov G. I., Karatsuba A. A., Chubarikov V. N., Teoriya kratnykh trigonometricheskikh summ, Nauka, M., 1987, 368 pp. | MR

[16] Arkhipov G. I., Chubarikov V. N., Karatsuba A. A., Trigonometric sums in number theory and analysis, De Gruyter expositions in mathematics, 39, Walter de Gruyter, Berlin–New York, 2004, 554 pp. | MR | Zbl

[17] Chubarikov V. N., “Kratnye trigonometricheskie summy s prostymi chislami”, Dokl. AN SSSR, 278:2 (1984), 302–304 | MR | Zbl

[18] Chubarikov V. N., “Otsenki kratnykh trigonometricheskikh summ s prostymi chislami”, Izv. AN SSSR, ser. matem., 49:5 (1985), 1031–1067 | MR | Zbl

[19] Leng S., Algebraicheskie chisla, Mir, M., 1966