© Young-Hee Kim et al. 2009
Received: 4 March 2009
Accepted: 25 April 2009
Published: 7 June 2009
for any elements , and distinct odd positive integers . We have the formulae for the complete sum of the products of -Euler polynomials related to the higher order -Euler polynomials using the fermionic -adic -Volkenborn integral on . We also obtain the formulae for the -Euler numbers.
In [21–24], Khrennikov introduced other theories of -adic distributions which were recently generated in -adic mathematical physics, both bosonic and fermionic: Khrennikov tried to build a -adic picture of reality based on the field of -adic numbers and corresponding analysis (a particular case of so-called non-Archimedean analysis). He showed that many problems of the description of reality with the aid of real numbers are induced by unlimited application of the Archimedean axiom. This axiom means that the physical observation can be measured with an infinite exactness. The results connected with an infinite exactness of measurements appear all the time in the formalisms of quantum mechanics and quantum field theories, which have the real continuum as one of their foundations. In particular, the author explains that the famous EPR paradox is nothing other than a result of using ideal real elements corresponding to an infinite exactness of measurement of the position and the momentum of a quantum particle. From the author's point of view, the EPR paradox is only a new form of Zeno's ancient paradox of Achilles and the tortoise. Both of these paradoxes are connected with the notion of an infinitely deep and infinitely divisible real continuum (see [21, 22]). In [23, 24], Khrennikov outlines both the -adic frequency model and a measure-theoretic approach. The latter is understood in the sense of non-Archimedean integration theory where measures have only additive property, not -additive property, and satisfy a condition of the boundedness. Analogues of the laws of large numbers including the central limit theorem are given. They studied a possible statistical interpretation of group-valued probabilities as well as nontraditional probabilistic models in physics and the cognitive sciences.
Therefore we have the following theorem.
By using the multinomial theorem, we can obtain the following Theorem 2.2. Theorem 2.2 is important to derive the main results of our paper.
By Theorem 2.2, we obtain
Hence we have the complete sum for q-Euler polynomials as follows.
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