An electrostatic model for zeros of classical Laguerre polynomials perturbed by a rational factor

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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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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Effect of delocalization of nonbonding electron density on the stability of the M–Ccarbene bond in main group metal-imidazol-2-ylidene complexes: a computational and structural database study

Physical Sciences Reviews ◽

10.1515/psr-2020-0084 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Samuel Tetteh ◽

Albert Ofori

Keyword(s):

Electron Density ◽

Lone Pair ◽

Main Group ◽

Binding Energies ◽

Electrostatic Model ◽

Group Metal ◽

Main Group Metal ◽

Structural Database ◽

The Stability ◽

Lone Pair Electrons

Abstract The M–Ccarbene bond in metal (M) complexes involving the imidazol-2-ylidene (Im) ligand has largely been described using the σ-donor only model with donation of σ electrons from the sp-hybridized orbital of the carbene carbon into vacant orbitals on the metal centre. Analyses of the M–Ccarbene bond in a series of group IA, IIA and IIIA main group metal complexes show that the M-Im interactions are mostly electrostatic with the M–Ccarbene bond distances greater than the sum of the respective covalent radii. Estimation of the binding energies of a series of metal hydride/fluoride/chloride imidazol-2-ylidene complexes revealed that the stability of the M–Ccarbene bond in these complexes is not always commensurate with the σ-only electrostatic model. Further natural bond orbital (NBO) analyses at the DFT/B3LYP level of theory revealed substantial covalency in the M–Ccarbene bond with minor delocalization of electron density from the lone pair electrons on the halide ligands into antibonding molecular orbitals on the Im ligand. Calculation of the thermodynamic stability of the M–Ccarbene bond showed that these interactions are mostly endothermic in the gas phase with reduced entropies giving an overall ΔG > 0.

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