An Exponential Discrete Convolution Model: Applications and Consequences

Let be a power function fr,M(s) defined for every s within the finite set M as follows (sr, s ∈ M, fr,M(s) = 0, otherwise. Let a discrete convolution of fr,M(s) be denoted as follows Convr,M[n] = (fr,M ∗ fr,M)[n]. Let a real coefficients Am,j be given by the following recurrence 0, if j < 0 or j > m,  , if 0 ≤ j < m, if j = m. In this paper we show that for every n > 0 the following odd-power identities involving coefficients Am,j and convolution transform Convr,M[n] hold Convr,N[n], n2m+1 − 1 = XAm,rConvr,Z>0[n], . 1. Definitions N - set of natural numbers {0,1,2,3,...}. Z>0 - set of positive integers {1,2,3,4,...}. Convr,M[n] = (fr,M ∗fr,M)[n] = Pk fr,M(k)fr,M(n−k) - convolution transform of real function fr,M(k) to itself. 2. Introduction and Main results The problem of finding expansions of monomials, binomials etc. is classical and there are a lot of beautiful solutions have been found, the most prominent examples are Binomial Theorem [1], Multinomial Theorem [7], Faulhaber's Formula [2], Worpitzky Identity [3], Identity in terms of Stirling numbers of the second kind and falling factorial [4]. Also, the 2010 Mathematics Subject Classification. 11C08 (primary), 44A35 (secondary). Key words and phrases. Power Identities, Polynomials, Convolution, Convolution power, Integral transforms. 1 one good example can be found at [6], so-called MacMillan Double binomial sum. Over decades mathematicians fight against the problem of polynomial expansions and this fight is successful, but still can we find some new approaches to solve this problem? This question is entire motivation of this manuscript. In this paper we solve the classical problem of finding expansions of monomials using convolution transform of power function, which defined on the finite set M. Let a power function fr,M be defined as follows . Mainly, we assume that the set M is set of natural numbers N or nonnegative integers Z>0. By this assumption it follows that convolution of fr,M(s) has a discrete form. Let the discrete convolution of fr,M(s) be defined as follows (2.1) Convr,M[n] := (fr,M ∗ fr,M)[n] = Xfr,M(k)fr,M(n − k). k If M is subset, but not a proper subset of N or Z>0, the formula (2.2) reduces to , if M ⊆N, (2.2) Conv , if M ⊆Z>0. Property 2.3. For every n,k fr,M(k)fr,M(n − k) = fr,M(n − k)fr,M(n − (n − k)) = fr,M(n − k)fr,M(k). Let a real coefficients Am,j be defined by the following recurrence relation Proposition 2.4. if j < 0 or j > m, , if 0 ≤ j < m, if j = m. Example of coefficients Am,j arranged in table Table 1. Coefficients Am,r. Note that the set of Am,j consists fractions for m ≥ 11. As Table 1 shows, for every m, the Am,0 = 1. AN ODD-POWER IDENTITY INVOLVING DISCRETE CONVOLUTION 3 The following theorem shows the odd-power identity involving coefficients Am,j and convolution transform Convr,M[n] = (fr,M ∗ fr,M)[n] Theorem 2.5. For every n,m ∈N m n Convr,N[n] = XAm,r Xkr(n − k)r, n > 0. r=0 k=0 As k approaches n in the sum, the kr(n − k)r takes nonzero value only in case when r = 0, , if k = n, r = 0; (2.6) if k = n, r > 0, we assume that there is 00 = 1 in (2.6). By the (2.6) and Theorem 2.5, Corollary 2.7. For every n,m ∈N m m n−1 n2m+1 − 1 = XAm,rConvr,Z>0[n] = XAm,r Xkr(n − k)r, n > 0. r=0 r=0 k=1 Corollary 2.8. For every n,m ∈N m n−1 n−1 m n2m+1 = XAm,r Xkr(n − k)r = XXAm,rkr(n − k)r. r=0 k=0 k=0 r=0 By the Property 2.3 (symmetry of kr(n − k)r), we also can rewrite Corollary 2.8 as n−1 m n m n2m+1 = XXAm,rkr(n − k)r = XXAm,rkr(n − k)r. k=0 r=0 k=1 r=0 One another interesting observation concerning the coefficients Am,r, the sum of Am,r over r gives (2.9) . Expression (2.9) is partial case of Corollary 2.8 for n = 2, it works since the for every r, the Convr,Z>0[2] = 1. 3. Proof of Theorem 2.5 Proof. By the Corollary 2.8, the coefficients Am,r could be evaluated expanding k)r and using Faulhaber's formula, we get (3.1) where Bs are Bernoulli numbers and. Now, we notice that if s = 0; , if s > 0. In particular, the last sum is zero for 0 < s ≤ r. Therefore, expression (3.1) takes the form Hence, introducing ` = 2r + 1 − s to (?) and ` = r − j to (), we get AN ODD-POWER IDENTITY INVOLVING DISCRETE CONVOLUTION 5 Using the definition of Am,r coefficients, we obtain the following identity for polynomials in n (3.2) Taking the coefficient of n2m+1 in (3.2) we get and taking the coefficient of n2d+1 for an integer d in the range m/2 ≤ d < m, we get Am,d = 0. Taking the coefficient of n2d+1 for d in the range m/4 ≤ d < m/2, we get , i.e, Continue similarly, we can express Am,d for each integer d in range m/2s+1 ≤ d < m/2s (iterating consecutively s = 1,2...) via previously determined values of Am,j as follows . Thus, for every (n,m) ∈N holds m n−1 n2m+1 = XAm,r Xkr(n − k)r r=0 k=0 By the (2.6), for every k = 0 or k = n in the convolution Convr,, the term kr(n − k)r equals to , if k = n, r = 0; if k = n, r > 0, Thus, Theorem 2.5, holds for every natural n > 0. This completes the proof. References Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 10, 1972. Knuth, D. E., Johann Faulhaber and Sums of Powers, pp. 9-10., arXiv preprint, arXiv:math/9207222v1 [math.CA], 1992. Worpitzky, J.. Studien uber die Bernoullischen und Eulerschen Zahlen.. Journal fur die reine und angewandte Mathematik 94 (1883): 203-232. http://eudml.org/doc/148532. Graham, Ronald, Knuth Donald, and Oren Patashnik (1989-01-05). Special Numbers. Concrete Mathematics (2nd ed.). Addison Wesley Longman Publishing Co. p. 262. ISBN 0-201-14236-8. D. V. Widder et al, "The Convolution Transform.", Bull. Amer. Math. Soc. 60 (1954), 444-456. https://doi.org/10.1090/S0002-9904-1954-09828-2. Weisstein, Eric W. "Power." From MathWorld–A Wolfram Web Resource, equation 12. Abramowitz, M. and Stegun, I. A. (Eds.). "Multinomial Coefficients." 24.1.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 823-824, 1972. URL: https://kolosovpetro.github.io

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