Resumen de Determinantal formulas for sum of generalized arithmetic-geometric series
- español
The main purpose of this paper is to give some closed form expressions by determinants for the sum of generalized arithmetic- geometric series. This will be done by solving the recurrence relation with combinatorial auto-convolution, satis ed by these sums. In a particular case, such a recurrence relation was obtained by L. Boul- ton and M. H. Rosas in this Boletin, [1]. Other recurrence relations of this type was solved by the author in [2] and [4], and was applied to study a new kind of equations - the di erential recurrence equa- tions with auto-convolution, both linear and combinatorial. The Catalan numbers also verify a recurrence relation with linear auto- convolution. See [5] or [7].
Applications to usual arithmetic-geometric series, sequences of con- sidered sums that are natural numbers, Fubini's numbers, Eulerian numbers and polynomials and some examples of Z-transforms are given. Another closed form expressions for the sum of generalized arithmetic- geometric series was given by R. Stalley, [10], using the properties of Stirling's numbers of second kind. A direct and ele- mentary proof for this result was given by the author in [3]
- English
The main purpose of this paper is to give some closed form expressions by determinants for the sum of generalized arithmetic- geometric series. This will be done by solving the recurrence relation with combinatorial auto-convolution, satis ed by these sums. In a particular case, such a recurrence relation was obtained by L. Boul- ton and M. H. Rosas in this Boletin, [1]. Other recurrence relations of this type was solved by the author in [2] and [4], and was applied to study a new kind of equations - the di erential recurrence equa- tions with auto-convolution, both linear and combinatorial. The Catalan numbers also verify a recurrence relation with linear auto- convolution. See [5] or [7].
Applications to usual arithmetic-geometric series, sequences of con- sidered sums that are natural numbers, Fubini's numbers, Eulerian numbers and polynomials and some examples of Z-transforms are given. Another closed form expressions for the sum of generalized arithmetic- geometric series was given by R. Stalley, [10], using the properties of Stirling's numbers of second kind. A direct and ele- mentary proof for this result was given by the author in [3]
