New $U$-Bernoulli, $U$-Euler and $U$-Genocchi polynomials and their matrices
William Ramírez, Daniel Bedoya, Alejandro Urieles, Clemente Cesarano, M.J. Ortega
Abstract
In this paper, we introduce the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomials, their numbers, and their relationship with the Riemann zeta function. We also derive the Apostol-type generalizations to obtain some of their algebraic and differential properties. We introduce generalized $U$-Bernoulli, $U$-Euler and $U$-Genocchi polynomial Pascal-type matrix. We deduce some product formulas related to this matrix. Furthermore, we establish some explicit expressions for the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomial matrices, which involves the generalized Pascal matrix.
Topics & Concepts
MathematicsBernoulli polynomialsProof of the Euler product formula for the Riemann zeta functionPascal (unit)Euler's formulaPure mathematicsBernoulli numberPascal matrixBernoulli's principleRiemann hypothesisOrthogonal polynomialsDifference polynomialsArithmetic zeta functionMathematical analysisMatrix functionSymmetric matrixPrime zeta functionQuantum mechanicsEngineeringAerospace engineeringEigenvalues and eigenvectorsPhysicsAdvanced Mathematical IdentitiesAdvanced Combinatorial MathematicsAlgebraic structures and combinatorial models