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In this paper we introduce different families of numerical and polynomial sequences by using Riordan pseudo involutions and Sheffer polynomial sequences. Many examples are given including dual of Hermite numbers and polynomials, dual of Bell numbers and polynomials, among other. The coefficients of some of these polynomials are related to the counting of different families of set partitions and permutations. We also studied the dual of Catalan numbers and dual of Fuss-Catalan numbers, giving several combinatorial identities.

This note presents a new formula of Eulerian numbers derived from Toeplitz matrices via Riordan array approach.

The familiar property of integral of f (x) = 1/x led to the discovery of other functions with this property.

Atherosclerotic plaques belong to the common vascular disease in the aged, which rupture will lead to acute thromboembolic diseases, the leading cause of fatal cardiovascular events. Accumulating evidence indicates that the lncRNAs-miRNAs-mRNA regulatory network plays a critical role in atherosclerosis. Based on RNA sequencing (GSE207252), we constructed expression profiles of lncRNAs, microRNAs, and mRNA in the carotid plaque of atherosclerosis patients and analyzed differentially expressed genes (DEGs). We identified three candidate lncRNAs using two algorithms (LASSO and SVM-RFE): lnc_GLRX3, lnc_FGF7-5, and DISC1FP1). LNCipedia, TargetScan, and miRDB databases were used to predict target miRNAs of lncRNAs and target genes of miRNAs. Gene ontology (GO) functional annotation, Kyoto Encyclopedia of Genes and Genomes (KEGG) pathway analysis, and Gene Set Enrichment Analysis (GSEA) analysis of DEGs was carried out using the R package clusterProfiler. A PPI network was constructed using the STRING website and visualized by Cytoscape. According to the "MCC" method of the plug-in cytoHubba in Cytoscape, ERCC4 was the top hub gene of the PPI network. We constructed a lncRNA_FGF7-5/lncRNA_GLRX3-miR-2681-5p-ERCC4 regulatory network for carotid plaque using lncRNA-miRNA and miRNA-mRNA pairs. Next, lncRNA_FGF7-5 and lncRNA_GLRX3 targeted miR-2681-5p directly to upregulate ERCC4 expression. Silencing of lncRNA_FGF7-5 and lncRNA_GLRX3 promoted apoptosis and TP53 expression in HUVECs treated with ox-LDL; however, these effects were reversed by ERCC4-overexpression. Taken together, these findings indicated that lncRNA_FGF7-5 and lncRNA_GLRX3 together reduced atherosclerosis-induced apoptosis of HUVECs via targeting miR-2681-5p to increase ERCC4 expression, thereby preventing the formation of carotid plaque and finally inhibiting atherosclerosis progression. [ABSTRACT FROM AUTHOR]

This note discussed the computation of Fermat's pseudoprimes (Fermat's Psp). [ABSTRACT FROM AUTHOR]

We present here a general rule of construction of identities for recursive sequences by using sequence transformation techniques developed in [16]. Numerous identities are constructed, and many well known identities can be proved readily by using this unified rule. Various Catalan-like and Cassini-like identities are given for recursive number sequences and recursive polynomial sequences. Sets of identities for Diophantine quadruple are shown. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to determine the number of solutions of three-variable Frobenius-related problems and to find their solutions by using order reducing methods. Here, the order of a Frobenius-related problem means the number of variables appearing in the problem. We present two types of order reduction methods that can be applied to the problem of finding all nonnegative solutions of three-variable Frobenius-related problems. The first method is used to reduce the equation of order three from a three-variable Frobenius-related problem to be a system of equations with two fixed variables. The second method reduces the equation of order three into three equations of order two, for which an algorithm is designed with an interesting open problem on solutions left as a conjecture. [ABSTRACT FROM AUTHOR]

When the golden ratio and its conjugate are zeros to a polynomial, two of the coefficients are functions of the Fibonacci sequence in terms of the other coefficients, which characterize the polynomial completely. These functions are used to derive some Fn, Ln, and golden ratio identities. In many cases, this is generalized to the Lucas sequences Un and Vn, with an associated quadratic root pair. Horadam sequences are produced in the the series of linear and constant coefficients of the series of polynomials having ra and rb zeros when all of the other coefficients are equal. [ABSTRACT FROM AUTHOR]

to construct a class of identities for number sequences generated by linear recur-rence relations. An alternative method based on the generating functions of the sequences is given. The equivalence between two methods for linear recurring sequences are also shown. However, the second method is not limited to the linear recurring sequences, which can be used for a wide class of sequences possessing rational generating functions. As examples, Many new and known identities of Stirling numbers of the second kind, Pell numbers, Jacob-sthal numbers, etc., are constructed by using our approach. Finally, we discuss the hyperbolic expression of the identities of linear recurring sequences. [ABSTRACT FROM AUTHOR]

Abstract: A relationship between a pair of Laurent series and Riordan arrays is formulated. In addition, a type of generalized Sheffer groups is defined by using Riordan arrays with respect to power series with non-zero coefficients. The isomorphism between a generalized Sheffer group and the group of the Riordan arrays associated with Laurent series is established. Furthermore, Appell, associated, Bell, and hitting-time subgroups of the groups are defined and discussed. A relationship between the generalized Sheffer groups with respect to different type of power series is presented. The equivalence of the defined Riordan array pairs and generalized Stirling number pairs is given. A type of inverse relations of various series is constructed by using pairs of Riordan arrays. Finally, several applications involving various arrays, polynomial sequences, special formulas and identities are also presented as illustrative examples. [Copyright &y& Elsevier]

Abstract: We define the Sheffer group of all Sheffer-type polynomials and prove the isomorphism between the Sheffer group and the Riordan group. An equivalence of the Riordan array pair and generalized Stirling number pair is also presented. Finally, we discuss a higher dimensional extension of Riordan array pairs. [Copyright &y& Elsevier]

Abstract: In this short paper, we give several new formulas for when n is an odd positive integer. The method is based on a recent proof, due to H. Tsumura, of Euler''s classical result for even n. Our results illuminate the similarities between the even and odd cases, and may give some insight into why the odd case is much more difficult. [Copyright &y& Elsevier]

Abstract: This paper deals with the convergence of the summation of power series of the form Σα≤κ(κ)χκ, where 0 ≤ α ≤ b ≤ ∞, and { (κ)} is a given sequence of numbers with κ ε (α, b) or(t) a differentiable function defined on (a, b). Here, the summation is found by using the symbolic operator approach shown in [1]. We will give a different type of the remainder of the summation formulas. The convergence of the corresponding power series will be determined consequently. Several examples such as the generalized Euler''s transformation series will also be given. In addition, we will compare the convergence of the given series transforms. [Copyright &y& Elsevier]

Abstract: Let φ be an orthonormal scaling function with approximation degree p − 1, and let Bn be the B-spline of order n. Then, spline type scaling functions defined by ƒ n = ƒ * Bn (n = 1, 2, ...) possess higher approximation order, p + n − 1, and compact support. The corresponding biorthogonal wavelet functions are also constructed. This technique is extended to the case of biorthogonal scaling function system. As an application of the method supplied in this paper, one can easily construct a sequence of pairs of biorthogonal spline type scaling functions from one pair of biorthogonal scaling functions or an orthonormal scaling function. In particular, if both the method and the lifting scheme of Sweldens (see [1]) are applied, then all pairs of biorthogonal spline type scaling functions shown in references [2] and [3] can be constructed from the Haar scaling function. [Copyright &y& Elsevier]

We define two type B -sequences of Riordan arrays and present the A -sequence characterization and Z -sequence characterization of the Riordan matrices with two type B -sequences. The subgroups characterized by A -sequences and Z -sequences are studied. The application of the sequence characterization to the RNA type matrices is discussed. Finally, we investigate the A - and B -sequences of the Pascal like Riordan matrices. [ABSTRACT FROM AUTHOR]