Estimating the polygamma functions for negative integers
© Salem and Kiliçman; licensee Springer. 2013
Received: 25 September 2013
Accepted: 25 September 2013
Published: 9 November 2013
The polygamma functions are defined for all and . In this paper, the concepts of neutrix and neutrix limit are applied to generalize and redefine the polygamma functions for all and . Also, further results are given.
Neutrices are additive groups of negligible functions that do not contain any constants except zero. Their calculus was developed by van der Corput  and Hadamard in connection with asymptotic series and divergent integrals. Recently, the concepts of neutrix and neutrix limit have been used widely in many applications in mathematics, physics and statistics.
For example, Jack Ng and van Dam applied the neutrix calculus, in conjunction with the Hadamard integral developed by van der Corput, to the quantum field theories, in particular, to obtain finite results for the coefficients in the perturbation series. They also applied the neutrix calculus to quantum field theory and obtained finite renormalization in the loop calculations, see  and .
Ozcag et al. [9, 10] applied the neutrix limit to extend the definition of incomplete beta function and its partial derivatives for negative integers. Also, the digamma function was generalized for negative integers by Jolevska-Tuneska et al. . Salem [12, 13] applied the neutrix limit to redefine the q-gamma and the incomplete q-gamma functions and their derivatives. In continuation, in this paper, we apply the concepts of neutrix and neutrix limit to generalize and redefine the polygamma functions.
A neutrix N is defined as a commutative additive group of functions defined on a domain with values in an additive group , where further if for some f in N, for all , then . The functions in N are called negligible functions.
Let N be a set contained in a topological space with a limit point a which does not belong to N. If is a function defined on with values in and it is possible to find a constant c such that , then c is called the neutrix limit of f as ξ tends to a, and we write .
Note that if a neutrix limit c exists, then it is unique, since if and are in N, then the constant function is also in N and so .
and all functions which converge to zero in the normal sense as ϵ tends to zero .
2 Gamma and digamma functions
where γ denotes Euler’s constant, see .
where γ is also Euler’s constant.
which is correct for all ; however, if , then is undefined, and we will give the exact formula for (2.12) in the following section.
3 Polygamma functions
In the present section, we seek to redefine the polygamma functions for all and .
Notice that the sum consists of (linear sum of ) negligible functions which can be neglected when taking the neutrix limit as . Taking the neutrix limit as yields the desired results. □
exists for all values of .
exist for all and thus integral (3.4) also exists. □
The results obtained in Lemma 3.1 and definition (3.1) complete the proof. □
The above theorems lead us to introducing the following.
for all and .
Remark 3.8 Formula (3.13) is the correct formula of (2.12).
The authors would like to thank the referee(s) for valuable remarks and suggestions on the previous version of the manuscript.
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