Identities associated with Milne-Thomson type polynomials and special numbers

Area of Science:

• Number Theory
• Combinatorics
• Mathematical Analysis

Background:

• Bernoulli numbers, Euler numbers, Stirling numbers, and Cauchy numbers are fundamental in number theory and combinatorics.
• Milne-Thomson and Hermite polynomials have significant applications in various mathematical fields.
• p-adic analysis provides powerful tools for studying number-theoretic objects.

Purpose of the Study:

• To establish new identities and relations involving a range of classical polynomials and number sequences.
• To explore the application of fermionic and bosonic p-adic integrals in deriving novel mathematical formulas.
• To connect different areas of mathematics through the study of combinatorial sums and special functions.

Main Methods:

• Utilizing fermionic and bosonic p-adic integrals as the primary analytical tools.
• Applying techniques from p-adic analysis to derive mathematical identities.
• Investigating combinatorial sums and their relationships with special polynomials and numbers.

Main Results:

• Derivation of novel identities and relations for Milne-Thomson polynomials, Hermite polynomials, Bernoulli numbers, Euler numbers, Stirling numbers, central factorial numbers, and Cauchy numbers.
• Establishment of new formulas connecting these mathematical entities through p-adic integration.
• Uncovering new relationships within combinatorial sums.

Conclusions:

• The study successfully extends the known properties of classical polynomials and numbers.
• Fermionic and bosonic p-adic integrals offer a potent framework for discovering new mathematical relationships.
• The findings contribute to a deeper understanding of the interconnections between number theory, combinatorics, and analysis.