We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wick’s theorem. The methodology can be straightforwardly generalized from the simple example we discuss to a wide class of operators.

We study generalized Cattaneo (telegrapher's) equations involving memory effects introduced by smearing the time derivatives. Consistency conditions where the smearing functions obey restrict freedom in their choice but the proposed scheme goes beyond the approach based on using fractional derivatives. We find conditions under which solutions of the equations considered so far can be recognized as probability distributions, i.e., are normalizable and nonnegative on their domains. Nonnegativity of solutions is demonstrated by methods of positive definite and completely monotonic functions with the Bernstein theorem being the cornerstone of the ongoing proofs. Analysis of exactly solvable examples and relevant mean-squared displacements enables us to classify diffusion processes described by such got solutions and to identify them with either ordinary or anomalous diffusion which character may change over time. To complete the present research we compare our results with those obtained using the continuous-time random-walk and the continuous-time persistent random-walk approaches.

We study functions related to the experimentally observed Havriliak–Negami dielectric relaxation pattern proportional in the frequency domain to [ 1 + ( i ω τ 0 ) α ] − β with τ 0 > 0 being some characteristic time. For α = l / k < 1 ( l and k being positive and relatively prime integers) and β > 0 we furnish exact and explicit expressions for response and relaxation functions in the time domain and suitable probability densities in their domain dual in the sense of the inverse Laplace transform. All these functions are expressed as finite sums of generalized hypergeometric functions, convenient to handle analytically and numerically. Introducing a reparameterization β = ( 2 − q ) / ( q − 1 ) and τ 0 = ( q − 1 ) 1 / α ( 1 < q < 2 ) we show that for 0 < α < 1 the response functions f α , β ( t / τ 0 ) go to the one-sided Lévy stable distributions when q tends to one. Moreover, applying the self-similarity property of the probability densities g α , β ( u ) , we introduce two-variable densities and show that they satisfy the integral form of the evolution equation.

Using the Bernstein theorem we prove the complete monotonicity of the three parameter Mittag?Leffler function E??,? (?w) for w ? 0 and suitably constrained parameters ?, ? and ?.