| Abstrakt: |
We study the sequence f n of functions that is defined by recursive convolutions as f 1 (x) = Π (x) , f n + 1 (x) = (f n ∗ f 1) (x) , n ∈ N , ![]() where Π is the unit rectangle function. We find out the general closed-form of the sequence f n and apply it for the evaluation of the improper integral 2 π ∫ 0 ∞ sin (ξ) ξ n cos (2 x ξ) d ξ , x ∈ R , n ∈ N , n ≥ 2. ![]() We also study some interesting features of the numerical coefficients that appear in the closed-form expression of f n . In connection to the numerical coefficients that appear in closed-form expression of f n , we introduce a map F defined on N × N 0 , by the rule, F (n , s) = ∑ i = 0 n i s (- 1) n - i i ! (n - i) ! , and show that its range is in N 0 , where N 0 : = N ∪ { 0 } . [ABSTRACT FROM AUTHOR] |
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