On the sum of the values of a polynomial at natural numbers which form a decreasing arithmetic progression

Bakir Farhi
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Bakir Farhi: Université de Bejaia

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 1, 15-27

Abstract: Abstract The purpose of this paper consists to study the sums of the type $$P(n) + P(n - d) + P(n - 2 d) + \dots $$ P ( n ) + P ( n - d ) + P ( n - 2 d ) + ⋯ , where P is a real polynomial, d is a positive integer and the sum stops at the value of P at the smallest natural number of the form $$(n - k d)$$ ( n - k d ) ( $$k \in {{\mathbb {N}}}$$ k ∈ N ). Precisely, for a given d, we characterize the $${{\mathbb {R}}}$$ R -vector space $${\mathscr {E}}_d$$ E d constituting of the real polynomials P for which the above sum is polynomial in n. The case $$d = 2$$ d = 2 is studied in more details. In the last part of the paper, we approach the problem through formal power series; this inspires us to generalize the spaces $${\mathscr {E}}_d$$ E d and the underlying results. Also, it should be pointed out that the paper is motivated by the curious formula: $$n^2 + (n - 2)^2 + (n - 4)^2 + \dots = \frac{n (n + 1) (n + 2)}{6}$$ n 2 + ( n - 2 ) 2 + ( n - 4 ) 2 + ⋯ = n ( n + 1 ) ( n + 2 ) 6 , due to Ibn al-Banna al-Marrakushi (around 1290).

Keywords: Ibn al-Banna’s formula; Bernoulli polynomials; Bernoulli numbers; Euler polynomials; Genocchi numbers; Primary 11B68; 11C08; 13F25 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00225-w

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