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Peer-reviewedCC BY 4.0Mathematics

Generalized Morgan-Voyce Polynomials with Positive and Negative Indices: Expansions and Analytic Properties

  • Waleed Mohamed Abd-Elhameed1
  • Omar Mazen Alqubori2
  • Naher Mohammed A. Alsafri2
  • Mohammed Atef Abdallah3
  • Amr Kamel Amin4,5
5 affiliations
  1. 1Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
  2. 2Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
  3. 3Department of Mathematics, Al-Lith University College, Umm Al-Qura University, Al-Lith 24382, Saudi Arabia
  4. 4Department of Mathematics, Adham University College, Umm Al-Qura University, Makkah 28653, Saudi Arabia
  5. 5Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef 62514, Egypt
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Abstract

This work develops a unified collection of algebraic and analytic identities for generalized Morgan-Voyce polynomials (GMVPs), including an explicit treatment of negative indices. Although negative-subscript Morgan-Voyce-type sequences have been considered previously, a unified treatment of the present one-parameter GMVP family that combines explicit positive- and negative-index connections with moment, inversion, linearization, differentiation, and integral identities does not appear to have been developed. Accordingly, the present work addresses this gap by establishing the structure of the negative-index GMVPs and their explicit connection with the ordinary nonnegative-index family. A moment representation and two inversion formulas are established, and the negative-index GMVPs are explicitly related to the ordinary nonnegative-index family. Linearization formulas and arbitrary-order derivative expansions are then derived, together with converse representations involving several auxiliary polynomial families. Closed-forms for some definite and weighted definite integrals are obtained. The resulting coefficients in several cases are found through terminating generalized hypergeometric series, which can be further simplified for particular choices of the involved parameters. The two classical Morgan-Voyce polynomial families are recovered as special cases of the generalized class, thereby placing their moment, inversion, connection, differentiation, linearization, and integration properties within a single unified framework.

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Cite

BibTeXRIS
BibTeX
@article{abdelhameed2026generalized,
  title = {{Generalized Morgan-Voyce Polynomials with Positive and Negative Indices: Expansions and Analytic Properties}},
  author = {Waleed Mohamed Abd-Elhameed and Omar Mazen Alqubori and Naher Mohammed A. Alsafri and Mohammed Atef Abdallah and Amr Kamel Amin},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14213889},
  url = {https://www.mdpi.com/2227-7390/14/21/3889},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Generalized Morgan-Voyce Polynomials with Positive and Negative Indices: Expansions and Analytic Properties
AU  - Waleed Mohamed Abd-Elhameed
AU  - Omar Mazen Alqubori
AU  - Naher Mohammed A. Alsafri
AU  - Mohammed Atef Abdallah
AU  - Amr Kamel Amin
PY  - 2026
DA  - 2026/10/25
JO  - Mathematics
DO  - 10.3390/math14213889
UR  - https://www.mdpi.com/2227-7390/14/21/3889
AB  - This work develops a unified collection of algebraic and analytic identities for generalized Morgan-Voyce polynomials (GMVPs), including an explicit treatment of negative indices. Although negative-subscript Morgan-Voyce-type sequences have been considered previously, a unified treatment of the present one-parameter GMVP family that combines explicit positive- and negative-index connections with moment, inversion, linearization, differentiation, and integral identities does not appear to have been developed. Accordingly, the present work addresses this gap by establishing the structure of the negative-index GMVPs and their explicit connection with the ordinary nonnegative-index family. A moment representation and two inversion formulas are established, and the negative-index GMVPs are explicitly related to the ordinary nonnegative-index family. Linearization formulas and arbitrary-order derivative expansions are then derived, together with converse representations involving several auxiliary polynomial families. Closed-forms for some definite and weighted definite integrals are obtained. The resulting coefficients in several cases are found through terminating generalized hypergeometric series, which can be further simplified for particular choices of the involved parameters. The two classical Morgan-Voyce polynomial families are recovered as special cases of the generalized class, thereby placing their moment, inversion, connection, differentiation, linearization, and integration properties within a single unified framework.
SN  - 2227-7390
ER  -