We denote by I ν and K ν the Bessel functions of the first and third kind, respectively. Motivated by the relevance of the function w ν t t I ν−1 t /I ν t , t > 0, in many contexts of applied mathematics and, in particular, in some elasticity problems Simpson and Spector 1984 , we establish new inequalities for I ν t /I ν−1 t. The results are based on the recurrence relations for I ν and I ν−1 and the Turán-type inequalities for such functions. Similar investigations are developed to establish new inequalities for K ν t /K ν−1 t .
A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951).
We first introduce a generalization of the Bernoulli polynomials, and consequently of the Bernoulli numbers, starting from suitable generating functions related to a class of Mittag-Leffler functions. Furthermore, multidimensional extensions of the Bernoulli and Appell polynomials are derived generalizing the relevant generating functions, and using the Hermite-Kampé de Fériet (or Gould-Hopper) polynomials. The main properties of these polynomial sets are shown. In particular, the differential equations can be constructed by means of the factorization method.
Abstract. We prove some monotonicity results for the incomplete gamma function,from which some inequalities for Γ(a, x) follow.Mathematics subject classification (1991): 33B15.
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