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In this paper, we present effective and efficient identification methods with a higher accuracy for fractional order system (FOS) with time delays and nonzero initial condition. We construct a new hybrid of orthonormal Bernoulli polynomials and block pulse functions (HOBPBPFs), and then derive the explicit representation for fractional integral operational matrix (FIOM) based on the HOBPBPFs via Laplace transformation. The orthogonality of polynomials gives us a higher accuracy of the proposed method. Based on it, new identification methods for FOS with input or state delay as well as nonzero initial condition are proposed. In particular, the variable least squares identification method for FOS with state delay using a hybrid seems to be the first within our knowledge. The effectiveness and efficiency of the proposed method is illustrated in several simulations via the comparison with methods using block pulse functions (BPFs) or a hybrid of Bernoulli polynomials and BPFs. [ABSTRACT FROM AUTHOR]

AbstractThe hyperbolic secant distribution has several generalizations with applications in, for example, finance. In this study, we explore the dual geometric structure of one such generalization: the beta-logistic distribution. Within this family, two special cases of random variables, as examples, are of particular interests: their moments, by some recent results, give the Bernoulli and Euler polynomials, which are important objects in many areas of mathematics. This current study also uncovers that the beta-logistic distribution admits a α-parallel prior for any real number α, that has the potential for application in geometric statistical inference. [ABSTRACT FROM AUTHOR]

In this paper, we present results on the uniqueness of the real zeros of the Hurwitz zeta function in given intervals. The uniqueness in question, if the zeros exist, has already been proved for the intervals (0 , 1) and (− N , − N + 1) for N ≥ 5 by Endo-Suzuki and Matsusaka, respectively. We prove the uniqueness of the real zeros in the remaining intervals by examining the behavior of certain associated polynomials. [ABSTRACT FROM AUTHOR]

In this paper, a combination approach based on Bernoulli polynomials and Gauss-Jacobi quadrature formula is developed to solve the system of nonlinear variable-order fractional Volterra integral equations (V-O-FVIEs). For this, we extend the constant coefficient in the Gauss-Jacobi formula to the variable coefficient and used it in our method. The method converts the system of V-O-FVIEs into the corresponding nonlinear system of algebraic equations. In addition, we use Gronwall inequality and the collectively compact theory to prove the existence and uniqueness of the solution of the original equation and the approximate equation, respectively. The convergence analysis and the error estimation of proposed method are discussed. Finally, some numerical examples illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

In the paper, the authors simply review recent results of inequalities, monotonicity, signs of determinants, determinantal formulas, closed-form expressions, and identities of the Bernoulli numbers and polynomials, establish an identity involving the differences between the Bernoulli polynomials and the Bernoulli numbers, present two identities among the differences between the Bernoulli polynomials and the Bernoulli numbers in terms of a determinant and a partial Bell polynomial, and derive a determinantal formula of the differences between the Bernoulli polynomials and the Bernoulli numbers.

In a previous work we studied generalized Stirling numbers of the second kind S(a2,b2,p2) a1,b1 (p1, k), where a1, a2, b1, b2 are given complex numbers, a1, a2 ?= 0, and p1, p2 are non-negative integers given. In this work we use these generalized Stirling numbers to define Bernoulli polynomials in two variables Bp1,p2 (x1, x2), and Euler polynomials in two variables Ep1,p2 (x1, x2). By using results for S(1,x2,p2) 1,x1 (p1, k), we obtain generalizations, to the bivariate case, of some well-known properties from the standard case, as addition formulas, difference equations and sums of powers. We obtain some identities for bivariate Bernoulli and Euler polynomials, and some generalizations, to the bivariate case, of several known identities for Bernoulli and Euler numbers and polynomials of the standard case. [ABSTRACT FROM AUTHOR]

This article, published in Mathematical Notes, discusses lacunary recurrence relations with gaps of length 8 for the Bernoulli and Euler polynomials. The authors use standard notations for these polynomials and numbers and provide formulas and expansions for them. They also mention the history of lacunary recurrence relations for these numbers and polynomials and highlight their own contributions in finding such relations. The authors describe the method of proof for their main result, Theorem 1, and provide corollaries that extend the relations to the Bernoulli and Euler numbers. The article concludes by acknowledging funding support and declaring no conflicts of interest. [Extracted from the article]

In the paper, the authors simply review recent results of inequalities, monotonicity, signs of determinants, determinantal formulas, closed-form expressions, and identities of the Bernoulli numbers and polynomials, establish an identity involving the differences between the Bernoulli polynomials and the Bernoulli numbers, present two identities among the differences between the Bernoulli polynomials and the Bernoulli numbers in terms of a determinant and a partial Bell polynomial, and derive a determinantal formula of the differences between the Bernoulli polynomials and the Bernoulli numbers. [ABSTRACT FROM AUTHOR]

The method of operational matrices based on different types of polynomials such as Bernstein, shifted Legendre and Bernoulli polynomials will be presented and implemented to solve the nonlinear Blasius equations approximately. The nonlinear differential equation will be converted into a system of nonlinear algebraic equations that can be solved using Mathematica®12. The efficiency of these methods has been studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as the polynomial degree (n) increases, since the errors decrease. Moreover, the approximate solutions obtained by the proposed methods are compared with the solution of the 4th order Runge-Kutta method (RK4), which gives very good agreement. In addition, the convergence of the proposed approximate methods is given based on one of the Banach fixed point theorem results.

The method of operational matrices is based on the Bernoulli and Shifted Legendre polynomials which is used to solve the Falkner-Skan equation. The nonlinear differential equation converting to a system of nonlinear equations is solved using Mathematica®12, and the approximate solutions are obtained. The efficiency of these methods was studied by calculating the maximum error remainder (MERm), and it was found that their efficiency increases as m increases. Moreover, the obtained approximate solutions are compared with the numerical solution obtained by the fourth-order Runge-Kutta method (RK4), which gives a good agreement. [ABSTRACT FROM AUTHOR]

This paper reports an efficient numerical method based on the Bernoulli polynomials for solving the variable-order fractional partial integro-differential equation(V-O-FPIDEs). The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann–Liouville integral operator are used in our paper. First, we prove the existence and uniqueness of the original equation by Gronwall inequality. Second, we use two-dimensional Bernoulli polynomials to approximate the unknown function of the original equation, and use the Gauss–Jacobi quadrature formula to deal with the variable-order Caputo fractional derivative operator and Riemann–Liouville integral operator. Then, the original equation is transformed into the corresponding system of algebraic equations. Furthermore, we give the convergence analysis and error estimation of the proposed method. Finally, some numerical examples illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

We systematically investigate the expressions and congruences for both a one-parameter family {Gn(x)} as well as a two-parameter family {Gn(r, m)} of sequences. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate a new subclass of biunivalent functions associated with the Bernoulli polynomials, which satisfy subordination conditions defined in the open unit disc. For this new subclass, we obtain estimates for the Taylor-Maclaurin coeffcients /α2/, /α3/ and Fekete-Szegö inequality /α3 - µa2². [ABSTRACT FROM AUTHOR]

We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials. [ABSTRACT FROM AUTHOR]

Abstract Identities of symmetry in two variables for Bernoulli polynomials and power sums had been investigated by considering suitable symmetric identities. T. Kim used a completely different tool, namely the p-adic Volkenborn integrals, to find the same identities of symmetry in two variables. Not much later, it was observed that this p-adic approach can be generalized to the case of three variables and shown that it gives some new identities of symmetry even in the case of two variables upon specializing one of the three variables. In this paper, we generalize the results in three variables to those in an arbitrary number of variables in a suitable setting and illustrate our results with some examples.

We introduce a certain multiple Hurwitz zeta function as a generalization of the Mordell–Tornheim multiple zeta function, and study its analytic properties. In particular, we evaluate the values of the function and its first and second derivatives at non-positive integers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate new subclasses of bi-univalent functions with respect to the symmetric points in U=z∈C:z<1 defined by Bernoulli polynomials. We obtain upper bounds for Taylor–Maclaurin coefficients a2,a3 and Fekete–Szegö inequalities a3−μa22 for these new subclasses.

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius–Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds [ABSTRACT FROM AUTHOR]

Let $p>3$ be a prime and let $a$ be a rational $p$ -adic integer with $a\not \equiv 0\;(\text{mod}\;p)$. We evaluate $$\begin{eqnarray}\mathop{\sum }_{k=1}^{(p-1)/2}\frac{1}{k}\binom{a}{k}\binom{-1-a}{k}\quad \text{and}\quad \mathop{\sum }_{k=0}^{(p-1)/2}\frac{1}{2k-1}\binom{a}{k}\binom{-1-a}{k}\end{eqnarray}$$ modulo $p^{2}$ in terms of Bernoulli and Euler polynomials. [ABSTRACT FROM AUTHOR]

If f (x , y) is a real function satisfying y > 0 and ∑ r = 0 n − 1 f (x + r y , n y) = f (x , y) for n = 1 , 2 , 3 , ... , we say that f (x , y) is an invariant function. Many special functions including Bernoulli polynomials, Gamma function and Hurwitz zeta function are related to invariant functions. In this paper we systematically investigate the properties of invariant functions. [ABSTRACT FROM AUTHOR]

The connection is considered between integrals and series involving polygamma ψ (z) and zeta ζ (z, s) functions, Bernoulli Bn (z) and Euler En (z) polynomials, and Bernoulli Bn and Euler En numbers. [ABSTRACT FROM AUTHOR]

Recently, some authors introduced poly-Genocchi polynomials as an analogy to poly-Bernoulli polynomials. In this paper, we will consider three types of sums of products of poly-Genocchi functions and derive their Fourier expansions. In addition, we will express each of them in terms of Bernoulli functions. [ABSTRACT FROM AUTHOR]

In this paper, we will consider three types of sums of finite products of Bernoulli and Euler functions, and derive the Fourier series expansions of them. In addition, we will express each of them in terms of Bernoulli functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a certain multiple zeta function as a generalization of the Mordell-Tornheim zeta function and study its analytic properties. In particular, we reveal its behavior at non-positive arguments by a new method which may be useful in a more general setting. [ABSTRACT FROM AUTHOR]

We relate geometric polynomials and -Bernoulli polynomials with an integral representation, then obtain several properties of -Bernoulli polynomials. These results yield new identities for Bernoulli numbers. Moreover, we evaluate a Faulhaber-type summation in terms of -Bernoulli polynomials. Finally, we introduce poly--Bernoulli polynomials and numbers, then study some arithmetical and number theoretical properties of them. [ABSTRACT FROM AUTHOR]

The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, is generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet L series, to Eisenstein series, and to general Mellin transforms. The Hurwitz numbers H ˜ n occur in the Laurent expansion about the origin of a certain Weierstrass ℘ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general ℘ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence. Other results concern the Matter numbers of the equianharmonic case of the ℘ function. [ABSTRACT FROM AUTHOR]