Abstract
We consider summation of some finite and infinite functional p-adic series with factorials. In particular, we are interested in the infinite series which are convergent for all primes p, and have the same integer value for an integer argument. In this paper, we present rather large class of such p-adic functional series with integer coefficients which contain factorials. By recurrence relations, we constructed sequence of polynomials A k (n; x) which are a generator for a few other sequences also relevant to some problems in number theory and combinatorics.
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References
I. Ya. Arefeva, B. Dragovich and I. V. Volovich, “On the p-adic summability of the anharmonic oscillator,” Phys. Lett. B 200, 512–514 (1988).
B. Dragovich, “p-Adic perturbation series and adelic summability,” Phys. Lett. B 256(3,4), 392–39 (1991).
B. G. Dragovich, “Power series everywhere convergent on R and Q p,” J. Math. Phys. 34(3), 1143–1148 (1992); [arXiv:math-ph/0402037].
B. G. Dragovich, “On p-adic aspects of some perturbation series,” Theor. Math. Phys. 93(2), 1225–1231 (1993).
B. G. Dragovich, “Rational summation of p-adic series,” Theor. Math. Phys. 100(3), 1055–1064 (1994).
B. Dragovich, “On p-adic series in mathematical physics,” Proc. Steklov Inst.Math. 203, 255–270 (1994).
B. Dragovich, “On p-adic series with rational sums,” Scientific Review19-20, 97–104 (1996).
B. Dragovich, “On some p-adic series with factorials,” in p-Adic Functional Analysis, Lect. Notes Pure Appl.Math. 192, 95–105 (Marcel Dekker, 1997); [arXiv:math-ph/0402050].
B. Dragovich, “On p-adic power series,” in p-Adic Functional Analysis, Lect. Notes Pure Appl.Math. 207, 65–75 (Marcel Dekker, 1999); [arXiv:math-ph/0402051].
B. Dragovich, “On some finite sums with factorials,” Facta Universitatis: Ser. Math. Inform. 14, 1–10 (1999); [arXiv:math/0404487 [math.NT]].
M. de Gosson, B. Dragovich and A. Khrennikov, “Some p-adic differential equations,” in p-Adic Functional Analysis, Lect. Notes Pure Appl. Math. 222, 91–112 (Marcel Dekker, 2001); [arXiv:math-ph/0010023].
L. Brekke and P. G. O. Freund, “p-adic numbers in physics,” Phys. Rep. 233, 1–66 (1993).
V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Sci. Publ., Singapore, 1994).
B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich, “On p-adic mathematical physics,” p-Adic Numbers Ultr. Anal. Appl. 1(1), 1–17 (2009); [arXiv:0904.4205 [math-ph]].
B. Dragovich and A. Yu. Dragovich, “A p-adic model of DNA sequence and genetic code,” p-Adic Numbers Ultr. Anal. Appl. 1(1), 34–41 (2009); [arXiv:q-bio/0607018 [q-bio.GN]].
W. H. Schikhof, Ultrametric Calculus: An Introduction to p-Adic Analysis (Cambridge Univ. Press, Cambridge, 1984).
M. RamMurty and S. Sumner, “On the p-adic series Σ ∞ n=1 n k · n!,” in Number Theory, CRM Proc. Lecture Notes 36, 219–227 (Amer. Math. Soc., 2004).
P. K. Saikia and D. Subedi, “Bell numbers, determinants and series,” Proc. Indian Acad. Sci. (Math. Sci.) 123(2), 151–166 (2013).
D. Subedi, “Complementary Bell numbers and p-adic series,” J. Integer Seq. 17, 1–14 (2014).
N. C. Alexander, “Non-vanishing of Uppuluri-Carpenter numbers,” Preprint http://tinyurl.com/oo36das.
N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” https://oeis.org/.
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Dragovich, B., Misic, N.Z. p-Adic invariant summation of some p-adic functional series. P-Adic Num Ultrametr Anal Appl 6, 275–283 (2014). https://doi.org/10.1134/S2070046614040025
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DOI: https://doi.org/10.1134/S2070046614040025