The Euler – Maclaurin Formula for Rational Parallelotope
The problem of summation of functions of a discrete argument is one of the classical problems of the calculus of finite differences, for example, the sum of the sequence of power of natural numbers was computed by Bernoulli (1713), and his studies led to the development of several branches of combinatorial analysis. Euler (1733) and independently Maclaurin (1738) found a formula in which the required sum is expressed through derivatives and the integral of the given function. Its first rigorous proof was given by Jacobi (1834).
A natural summation of functions of several discrete arguments is over integer points of rational polytopes. The analogues of the Euler – Maclaurin formula for summation of polynomials over an arbitrary rational polytope and for summation of function of exponential type over integer points of simplex are known.
In this article we obtain a multidimensional analogue of the Euler-Maclaurin formula for summation of entire functions of exponential type over integer points of rational parallelotops built on generators of a unimodular rational cone. Unimodularity of the cone is essential since in the chosen method of proof it allows us to change variables and replace the parallelotope by the parallelepiped. Also, we implement Euler’s approach based on the concept of discrete primitive functions. Namely, using the methods of the theory of multidimensional difference equations, the concept of a generalized discrete primitive is introduced, and the methods of the theory of differential operators of infinite order allow to justify the convergence of series of functions that appear in the Euler – Maclaurin formula.
1. Gelfond A.O. Calculus of finite differences (in Russian). Moscow, Nauka, 1977.
2. Dubinsky Yu.A. The Cauchy problem in the complex domain (in Russian). Moscow, Izdatelstvo MEI, 1996.
3. Nekrasova T.I. Cauchy problem for multidimensional difference equations in lattice cones (in Russian). Journal of Siberian Federal University, Mathematics & Physics,;2012, vol. 5, issue 4, pp. 576-580.
4. Nekrasova T.I. Sufficient conditions of algebraicity of generating functions of the solutions of multidimensional difference equations (in Russian). Izvestiya Irkutskogo Gosudarstvennogo Universiteta. Seriya "Matematika", 2013, vol. 6, no 3, pp. 88-96.
5. Abramov S.A. On the summation of rational functions. USSR Comput. Math. Math. Phys., 1971, vol. 11, no 4, pp. 324-330.
6. Bousquet-M´elou M., Petcovˇsek M. Linear recurrences with constant coefficients: the multivariate case. Discrete Mathematics, 2000, vol.225, pp. 51-75.
7. Brion M., Berline N. Local Euler – Maclaurin formula for polytopes. Moscow Mathematical Society Journal, 2007, vol. 7, pp. 355-383.
8. Brion M., Vergne M. Residue formulae, vector partition functions and lattice points in rational polytopes. Journal of the American Mathematical Society, 1997, vol. 10, no 4, pp. 797-833.
9. Hardy G. Divergent series. London, Oxford University Press, 1949.
10. Leinartas E.K. Multiple Laurent series and fundamental solutions of linear difference equations. Siberian Mathematical Journal, 2007, vol.48, no 2, pp. 268-272.
11. Polyakov S.A. Indefinite summation of rational functions with factorization of denominators. Programming and Computer Software, 2011, vol. 37, no 6, pp. 322-325.
12. Shishkina O.A. The Euler-Maclaurin Formula and Differential Operators of Infinite Order. Journal of Siberian Federal University, Mathematics & Physics,;2015, vol. 8, no 1, pp. 86-93.