A Survey on p-Adic Integrals

The p-adic numbers are a counterintuitive arithmetic system and were firstly introduced circa end of the nineteenth century. In conjunction with the introduction of these numbers, many mathematicians and physicists started to develop new scientific tools using their available, useful, and applicable properties. Several effects of these researches have emerged in mathematics and physics such as p-adic analysis, string theory, p-adic quantum mechanics, quantum field theory, representation theory, algebraic geometry, complex systems, dynamical systems, and genetic codes. One of the important tools of the mentioned advancements is the p-adic integrals. Intense research activities in such an area like p-adic integrals are principally motivated by their significance in p-adic analysis. Recently, p-adic integrals and its diverse extensions have been studied and investigated in detail by many mathematicians. This chapter considers and investigates multifarious extensions of the p-adic integrals elaborately. q-Analogues with diverse extensions of p-adic integrals are also considered such as the weighted p-adic q-integral on ℤ p $$ \mathbb {Z} _{p}$$ . The two types of the weighted q-Boole polynomials and numbers are introduced and investigated in detail. As several special polynomials and numbers can be derived from the p-adic integrals, some generalized and classical q-polynomials and numbers are obtained from the aforesaid extensions of p-adic integrals. Finally, the importance of these extensions is analyzed.