Current implementations of p-adic numbers usually rely on so called zealous algorithms, which compute with truncated p-adic expansions at a precision that can be specified by the user. In combination with Newton-Hensel type lifting techniques, zealous algorithms can be made very efficient from an asymptotic point of view. In the similar context of formal power series, another so called lazy tec...

Approximation lattices occur in a natural way in the study of rational approximations to p-adic numbers. Periodicity of a sequence of approximation lattices is shown to occur for rational and quadratic p-adic numbers. and for those only, thus establishing a p-adic analogue of Lagrange’s theorem on periodic continued fractions. Using approximation lattices we derive upper and lower bounds for th...

Lengyel introduced a sequence of numbers Zn, defined combinatorially, that satisfy a recurrence where the coefficients are Stirling numbers of the second kind. He proved some 2-adic properties of these numbers. In this paper, we give another recurrence for the sequence Zn, where the coefficients are Stirling numbers of the first kind. Using this formula, we give another proof of Lengyel’s lower...

The purpose of this paper is to derive formulae for the sums of products of the q-Euler polynomials and numbers, since many identities can be obtained from our sums of products of the q-Euler polynomials and numbers. In 1 , Simsek evaluated the complete sums for the Euler numbers and polynomials and obtained some identities related to Euler numbers and polynomials from his complete sums, and Ja...

An effective p-adic encoding of dendrograms is presented through an explicit embedding into the Bruhat-Tits tree for a p-adic number field. This field depends on the number of children of a vertex and is a finite extension of the field of p-adic numbers. It is shown that fixing p-adic representatives of the residue field allows a natural way of encoding strings by identifying a given alphabet w...

Let p be a fixed odd prime number. Throughout this paper, Zp, Qp and Cp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and N = N ∪ {0}. The p-adic norm is normally defined by |p|p = 1/p. As an indeterminate, we assume that q ∈ Cp with |1 − q|p < 1 (see [1-43]...

This paper is the first one in the series devoted to the calculation of particle mass spectrum in Topological GeometroDynamics. In this paper p-adic conformal field theory limit of TGD is formulated. TGD Universe is critical at quantum level and the idea is to realize criticality via conformal invariance. Ordinary real numbers do not allow this but if one assumes that in long length scales p-ad...

Let p be a fixed odd prime number. Throughout this paper Zp,Qp, and Cp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp. Let N be the set of natural numbers and Z N ∪ {0}. The p-adic norm on Cp is normalized so that |p|p p−1. Let C Zp be the space of continuous functions on Zp. For f ∈ C Zp , the fermionic ...

Let p be a fixed odd prime number. Throughout this paper, Zp, Qp, and Cp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively. The p-adic norm is normalized so that |p|p 1/p. Let N be the set of natural numbers and Z N ∪ {0}. Let UD Zp be the space of uniformly differentiable functions on Zp. For f ∈ ...

Let p be a fixed odd prime number. Throughout this paper, Zp, Qp, and Cp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N ∪ {0}. The p-adic absolute value on Cp is normalized so that |p|p 1/p. Assume that q ∈ Cp with |1 − q|p < 1. Let f be a continuous ...