Congruence properties of Hermite and Laguerre polynomials

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.

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Algebras Describing Pseudocomplemented, Relatively Pseudocomplemented and Sectionally Pseudocomplemented Posets

Symmetry ◽

10.3390/sym13050753 ◽

2021 ◽

Vol 13 (5) ◽

pp. 753

Author(s):

Ivan Chajda ◽

Helmut Länger

Keyword(s):

Universal Algebra ◽

Relational Structures ◽

Pseudocomplemented Poset ◽

The Given ◽

The Relationship ◽

Congruence Properties

In order to be able to use methods of universal algebra for investigating posets, we assigned to every pseudocomplemented poset, to every relatively pseudocomplemented poset and to every sectionally pseudocomplemented poset, a certain algebra (based on a commutative directoid or on a λ-lattice) which satisfies certain identities and implications. We show that the assigned algebras fully characterize the given corresponding posets. A certain kind of symmetry can be seen in the relationship between the classes of mentioned posets and the classes of directoids and λ-lattices representing these relational structures. As we show in the paper, this relationship is fully symmetric. Our results show that the assigned algebras satisfy strong congruence properties which can be transferred back to the posets. We also mention applications of such posets in certain non-classical logics.

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A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Symmetry ◽

10.3390/sym13040648 ◽

2021 ◽

Vol 13 (4) ◽

pp. 648

Author(s):

Ghulam Muhiuddin ◽

Waseem Ahmad Khan ◽

Ugur Duran ◽

Deena Al-Kadi

Keyword(s):

Generating Function ◽

Functional Equations ◽

Laguerre Polynomials ◽

Hermite Polynomials ◽

Bernoulli Polynomials ◽

Higher Order ◽

New Class

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

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