Statistics

Lucas sequence

Fibonacci number

Stirling numbers of the second kind

Real number

Bell polynomials

Economics

Combinatorics

Classical orthogonal polynomials

Binomial coefficient

Stirling numbers of the first kind

Finance

Bernoulli number

Expected value

Pure mathematics

Mathematics

Catalan number

Stirling number

Order (exchange)

Lucas number

Recurrence relation

Orthogonal polynomials

Discrete mathematics

Fibonacci polynomials

Binomial (polynomial)

Harmonic number

Riemann zeta function

Applicable Analysis and Discrete Mathematics

The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to many well-known numbers, which are Bernoulli numbers, Fibonacci numbers, Lucas numbers, Stirling numbers of the second kind and central factorial numbers. Our other inspiration of this paper is related to the Golombek's problem [15] "Aufgabe 1088. El. Math., 49 (1994), 126-127". Our first numbers are not only related to the Golombek's problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by tables. We give some applications in probability and statistics. That is, special values of mathematical expectation of the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, we derive recurrence relations and some formulas of our numbers. Moreover, we come up with a conjecture with two open questions related to our new numbers. We give two algorithms for computation of our numbers. We also give some combinatorial applications, further remarks on our new numbers and their generating functions.

New families of special numbers for computing negative order Euler numbers and related numbers and polynomials