### $\mathbb{P}$-species and the $q$-Mehler formula.

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We give the following bounds on Laguerre polynomials and their derivatives (α ≥ 0): $\genfrac{}{}{0pt}{}{|{t}^{k}{d}^{p}\left({L}_{n}^{\alpha}\left(t\right){e}^{-t/2}\right)|\le {2}^{-min(\alpha ,k)}{4}^{k}(n+1)...(n+k)(n+p+max(\alpha -k,0)}{n)}$ for all natural numbers k, p, n ≥ 0 and t ≥ 0. Also, we give (as the main result of this paper) a technique to estimate the order in k and p in bounds similar to the previous ones, which will be used to see that the estimate on k and p in the previous bounds is sharp and to give an estimate on k and p in other bounds on the Laguerre polynomials proved by Szegö.

2000 Mathematics Subject Classification: 26A33, 33C60, 44A20In this survey we present a brief history and the basic ideas of the generalized fractional calculus (GFC). The notion “generalized operator of fractional integration” appeared in the papers of the jubilarian Prof. S.L. Kalla in the years 1969-1979 when he suggested the general form of these operators and studied examples of them whose kernels were special functions as the Gauss and generalized hypergeometric functions, including arbitrary...