  The main purpose of this paper is to provide a novel approach to deriving
formulas for the p-adic q-integral including the Volkenborn integral and the
p-adic fermionic integral. By applying integral equations and these integral
formulas to the falling factorials, the rising factorials and binomial
coefficients, we derive some new and old identities and relations related to
various combinatorial sums, well-known special numbers such as the Bernoulli
and Euler numbers, the harmonic numbers, the Stirling numbers, the Lah numbers,
the Harmonic numbers, the Fubini numbers, the Daehee numbers and the Changhee
numbers. Applying these identities and formulas, we give some new combinatorial
sums. Finally, by using integral equations, we derive generating functions for
new families of special numbers and polynomials. We also give further comments
and remarks on these functions, numbers and integral formulas.
