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In this paper, we study the generating functions of multiple t-star values with an arbitrary number of blocks of twos, which are based on the results of the corresponding generating functions of multiple t-harmonic star sums. These generating functions can deduce an explicit expression of multiple t-star values. As applications, we obtain some evaluations of multiple t-star values that represent the relations between multiple t-star values and alternating multiple t-half values. [ABSTRACT FROM AUTHOR]

As a structural topological index, the number of subtrees has great significance for the analysis and design of hybrid locally reliable networks. In this paper, with generating function and introducing a novel two-forest dual transformation technique, we solve the subtree enumerating problems of two representatives of the self-similar networks, such as the hierarchical lattice and |$(u,v)$| -flower networks. Moreover, by means of the circle weight transfer technique, two linear time algorithms of computing the subtree generation functions of these two families of networks are also proposed. The subtree density of two special cases for these self-similar networks is briefly discussed as an application. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of special polynomials called the generalized Bell polynomials, constructed by combining two-variable general polynomials with two-variable Bell polynomials. The concept of the monomiality principle was employed to establish the generating function and obtain various results for these polynomials. We explore certain related identities, properties, as well as differential and integral formulas. Further, specific members within the generalized Bell family—such as the Gould-Hopper-Bell polynomials, Laguerre-Bell polynomials, truncated-exponential-Bell polynomials, Hermite-Appell-Bell polynomials, and Fubini-Bell polynomials—were examined, unveiling analogous outcomes for each. Finally, Mathematica was utilized to investigate the zero distributions of the Gould-Hopper-Bell polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we define the bi-periodic k-Pell sequence. We obtain Binet's formula, some identities of the bi-periodic k-Pell sequences like Catalan, Cassini and D'Ocagne's identities as well as some related summation formulas. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the compositions of the integers. We present the decompositions for both the composition sets and the odd composition sets of the integers. Thus the decompositions provide us to have not only an alternative proof of some well known identies but also many new identities for Fibonacci numbers and Lucas numbers. Thus we investigate the generating functions for the product sum of the odd composition sets of the integers and attain some functional equations. [ABSTRACT FROM AUTHOR]

The Hybrid numbers are generalizations of complex, hyperbolic and dual numbers. In recent years, studies related with hybrid numbers have been increased significantly. In this paper, we introduce the generalized bivariate conditional Fibonacci and Lucas hybrinomials. Also, we present the Binet formula, generating functions, some significant identities, Catalan's identities and Cassini's identities of the generalized bivariate conditional Fibonacci and Lucas hybrinomials. Finally, we give more general results compared to the previous works. [ABSTRACT FROM AUTHOR]

In this paper, we fully solve the problem of the completeness of the eigenfunctions of an ordinary 5th-order differential operator in the space of square-summable functions on the segment [0, 1] generated by the simplest differential expression y(5) and two-point two-term boundary conditions ανy(ν−1)(0) + βνy(ν−1)(1) = 0 and v=1,5-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v=\stackrel{-}{\text{1,5}}$$\end{document} under the main assumption αν ≠ 0,v=1,5-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v=\stackrel{-}{\text{1,5}}$$\end{document} or βν ≠ 0, v=1,5-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v=\stackrel{-}{\text{1,5}}$$\end{document} (in this case, without loss of generality, we can assume that all αν or all βν, respectively, are equal to one).The classical methods of studying completeness, which go back to well-known articles by Keldysh, Khromov, Shkalikov, and many others, are not applicable to the operator under consideration. These methods are based on “good” estimates for the spectral parameter of the used generating functions (“classical”) for the system of eigenfunctions and associated functions. In the case of a strong irregularity of the operator under consideration, these “classical” generating functions have too large rate of grows in the spectral parameter. To solve the problem of multiple completeness, we propose a new approach that uses a special parametric solution that generalizes “classical” generating functions. The main idea of this approach is to select the parameters of this special solution to construct generating functions that are no longer “classical” with suitable estimates in terms of the spectral parameter. Such a selection for the operator under consideration turned out to be possible, although rather nontrivial, which allowed us to follow the traditional scheme of proving the completeness of the system of eigenfunctions in the space of square-summable functions on the segment [0, 1]. [ABSTRACT FROM AUTHOR]

An adjacent q -cycle is a natural generalization of an adjacent transposition. We show that the number of adjacent q -cycles in a permutation maps to the sum of occurrences of two mesh patterns under Foata's fundamental transformation. As a corollary we resolve Conjecture 3.14 in the paper "From Hertzprung's problem to pattern-rewriting systems" by the first author. [ABSTRACT FROM AUTHOR]

A multiple leaf-distance granular regular α -tree (abbreviated as LDR α -tree for short) is a tree (with at least α + 1 vertices) where any two leaves are at some distance divisible by α. A connected graph's subtree which is additionally an LDR α -tree is known as an LDR α -subtree. Obviously, α = 1 and 2, correspond to the general subtrees (excluding the single vertex subtrees) and the BC-subtrees (the distance between any two leaves of the subtree is even), respectively. With generating functions and structure decomposition, in this paper, we propose algorithms for enumerating an auxiliary subtree α τ (v) -subtree (τ = 0 , 1 , ... , α − 1) containing a fixed vertex, and various LDR α -subtrees of unicyclic graphs, respectively. Basing on these algorithms, we further present algorithms for enumerating various LDR α -subtrees of edge-disjoint bicyclic graphs. [ABSTRACT FROM AUTHOR]