Appl. Math. Inf. Sci. 7, No. 1, 167-170 (2013) 167 Applied Mathematics & Information Sciences An International Journal °c 2013 NSP Natural Sciences Publishing Cor. Some results on the digamma function Biljana Jolevska-Tuneska1 and Ilija Jolevski2 1 Faculty of Electrical Engineering and Informational Technologies, Karpos II bb, Skopje, Republic of Macedonia 2 Technical Faculty Bitola, University of St. Kliment Ohridski, Bitola, Republic of Macedonia Received: 19 Jul. 2011; Revised 11 Oct. 2012; Accepted 23 Oct. 2012 Published online: 1 Jan. 2012 Abstract: The digamma function is defined for x > 0 as a locally summable function on the real line by Z 1 e¡t ¡ e¡xt Ã(x) = ¡γ + ¡t dt : 0 1 ¡ e In this paper we use the neutrix calculus to extend the definition for digamma function for the negative integers. Also we consider the derivatives of the digamma function for negative integers. Keywords: Digamma Function, Gamma Function, Delta Function, Neutrices 1. Introduction defined infinite quantities was devised by Hadamard and the resulting finite value extracted from the divergent inte- The gamma function ¡ (x) was introduced by Leonard Eu- gral is usually referred to as Hadamard finite part. ler as a generalization of the factorial function on the set Using the concepts of the neutrix and the neutrix limit of all real numbers. It is defined for x > 0 by:(see [3] and due to van der Corput [4], Fisher gave the general princi- [13]) Z 1 ple for the discarding of unwanted infinite quantities from ¡ (x) = tx¡1e¡t dt : asymptotic expansions and has been exploited in context 0 of distributions, see [7,6]. Due to the difficulties in dealing with ¡ 0(x) in particulary We also note that recently Ng and van Dam applied because it is a large function that increases very rapidly, the neutrix calculus, in conjunction with the Hadamar in- the logarithmic derivative of ¡ (x) is studied instead. This tegral, developed by van der Corput, to the quantum field function is known as the digamma function (or psi func- theories, in particular to obtain finite results for the coeffi- tion) Ã(x) (see [5,2]) and is defined for positive real num- cients in the perturbation series. They also applied neutrix bers x by: calculus to quantum field theory, and obtained finite renor- malization in the loop calculations, see [16,17]. d [ln ¡ (x)] ¡ 0(x) Ã(x) = = : dx ¡ (x) 2. Neutrix calcucus The infinite family of approximations of the Digammma function were recently considered by I. Muqattash and In the following, we let N be the neutrix, see van der Cor- M.Yahdi (see [1]). put [4], having domain N 0 = f² : 0 < ² < 1g and In this paper we consider the values of the digamma negligible functions finite linear sums of the functions function for negative integers. Also we consider the deriva- ²¸ lnr¡1 ²; lnr ² : ¸ < 0; r = 1; 2;::: tives of the digamma function for negative integers. To define the digamma function for this values we use neu- and all functions f(²) which tend to zero in the normal trix calculus. The technique of neglecting appropriately sense as ² tends to zero. ¤ Corresponding author: e-mail: [email protected] °c 2013 NSP Natural Sciences Publishing Cor. 168 B. Jolevska-Tuneska et al : Some results on the digamma function If f(²) is a real (or complex) valued function defined for r = 0; 1; 2;::: and m = 1; 2;::: on N 0 and if it is possible to find a constant c such that In [10] and [8] Fisher et al. defined the incomplete f(²) ¡ c is in N, then c is called the neutrix limit of f(²) gamma function γ(¡m; x) for m = 0; 1; 2;::: Some con- as ² ! 0 ; and we write N¡lim f(²) = c : Note that taking volutions and neutrix convolution of this function and other ²!0 functions were considered in [11] and [12]. Recently, Ozcag the neutrix limit of a function f(²) is equivalent to taking et al. in [15] defined the incomplete beta function and its the usual limit of Hadamar’s finite part of f(²), and if a derivatives. function f(²) tends to c in the normal sense as ² tends to zero, also it converges to c in the neutrix sense. The reader may find the general definition of a neutrix limit with some 3. Main result examples in [9]. In the following we apply Fishers’s principle to define The digamma function (see [14]) has its integral represen- the digamma function for negative integers. tation Z It was proved in [9] that 1 e¡t ¡ e¡xt Z Ã(x) = ¡γ + ¡t dt ; 1 0 1 ¡ e ¡ (x) = N¡lim tx¡1e¡t dt ²!0 and the integral is convergent for x > 0 : Also, for x > 0 ; ² this can be written as: for x 6= 0; ¡1; ¡2;:::; where ¡ denotes the Gamma func- Z 1 1 ¡ tx¡1 tion. This suggested that ¡ (¡m) be defined by: Ã(x) = ¡γ + dt : (3) Z 1 0 1 ¡ t ¡ (¡m) = N¡lim t¡m¡1e¡t dt (1) It can be easily proved from equation (3) that: ²!0 ² 1 for m = 0; 1; 2;:::: Ã(x + 1) = Ã(x) + ; (4) It was also proved that x Z 1 for x > 0 and this equation can be used to define Ã(x) ¡ (0) = e¡t ln t dt = ¡ 0(1) = ¡γ; for negative, non-integer values of x. Thus if ¡1 < x < 0 0 then 1 where γ denotes Euler’s constant, and Ã(x) = Ã(x + 1) ¡ : Z 1 x ¡ (¡m) = t¡m¡1e¡t dt + So, we have that if ¡n < x < ¡n + 1 ; n = 1; 2;:::; 1 then: Z m 1 h X i i Z 1 x¡1+n n ¡m¡1 ¡t (¡t) 1 ¡ t X 1 + t e ¡ dt + Ã(x) = ¡γ + dt ¡ : (5) 0 i! 1 ¡ t x + k ¡ 1 i=0 0 k=1 m¡1 X (¡1)i It follows that if ¡n < x < ¡n + 1 ; n = 1; 2;:::; and ¡ (2) i!(m ¡ i) x > 0, then: i=0 Z Z 1 1 ¡ tx¡1 1 1 ¡ tx¡1+n for m = 0; 1; 2;:::: ¡γ + dt = ¡γ + dt ¡ Fisher and Kuribayashi [9] proved the existence of ² 1 ¡ t ² 1 ¡ t (r) (r) Z ¡ (0) and then defined ¡ (0) by the equation: 1 tx¡1(1 ¡ tn) Z 1 ¡ dt = ¡ (r)(0) = N¡lim t¡1 lnr te¡t dt ² 1 ¡ t Z n ²!0 ² 1 1 ¡ tx¡1+n X 1 Z 1 Z 1 = ¡γ + dt ¡ + ¡1 r ¡t ¡1 r ¡t 1 ¡ t x + k ¡ 1 = t ln te dt + t ln t[e ¡ 1] dt ; ² k=1 1 0 Xn ²x+k¡1 for r = 0; 1; 2;::: This suggested that ¡ (r)(¡m) be de- + ; fined by: x + k ¡ 1 Z k=1 1 and it follows that ¡ (r)(¡m) = N¡lim t¡m¡1 lnr te¡t dt Z 1 1 ¡ tx¡1 ²!0 ² N¡lim dt = Z 1 ²!0 ² 1 ¡ t = t¡m¡1 lnr te¡t dt + " # Z 1 x¡1+n n 1 1 ¡ t X 1 " # = lim dt ¡ + Z 1 m i X (¡t) ²!0 ² 1 ¡ t x + k ¡ 1 + t¡m¡1 lnr t e¡t ¡ dt + k=1 i! n x+k¡1 0 i=0 X ² + N¡lim = mX¡1 (¡1)i ²!0 x + k ¡ 1 + r!(m ¡ i)¡r¡1 ; k=1 i! i=0 = Ã(x) + γ °c 2013 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 7, No. 1, 167-170 (2013) / www.naturalspublishing.com/Journals.asp 169 on using equation (5). We have therefore shown that Definition 2The derivative of the digamma function Ã(x) Z is defined by 1 1 ¡ tx¡1 Ã(x) = ¡γ + N¡lim dt (6) Z 1 ²!0 1 ¡ t t¡n¡1 ln t ² Ã0(¡n) = N¡lim dt (9) for x 6= 0; ¡1; ¡2;:::: This suggest the following defini- ²!0 ² 1 ¡ t tion. for n = 0; 1; 2;:::; provided that the neutrix limit exists. Definition 1The digamma function Ã(x) is defined by Z Now we have the following theorem. 1 1 ¡ t¡n¡1 Ã(¡n) = ¡γ + N¡lim dt (7) Theorem 2.The derivative of the digamma function Ã(x) ²!0 ² 1 ¡ t have values for negative integers, and for n = 0; 1; 2;:::; provided that the neutrix limit exists. X1 1 Now we have the following theorem. Ã0(¡n) = : (10) (k ¡ n)2 Theorem 1.The digamma function Ã(x) have values for k=1 negative integers, and for n = 0; 1; 2;::: Xn 1 Ã(¡n) = ¡γ + : (8) Proof. Using the equation k Z Z k=1 1 t¡n¡1 ln t X1 1 ¡ dt = t¡n¡1+i ln t dt for n = 0; 1; 2;::: 1 ¡ t ² i=0 ² Proof. We will now prove the existence of Ã(0): We 1 · ¸ X ²¡n+i ln ² ²¡n+i ¡ 1 have Z Z = + ; 1 1 ¡ t¡1 1 dt ¡n + i (¡n + i)2 dt = ¡ = ¡ ln ² ; i=0 ² 1 ¡ t ² t and taking the neutrix limit we have that Z 1 ¡1 1 ¡ t Z and it follows that N¡lim dt = 0 :Ã(0) there- 1 t¡n¡1 ln t X1 1 ²!0 ² 1 ¡ t N¡lim dt = ; fore exists and Ã(0) = ¡γ : ²!0 1 ¡ t (k ¡ n)2 ² k=1 Next we have: Z 1 1 ¡ t¡n¡1 Xn 1 Xn 1 proving the theorem. dt = ¡ + ln ² ; 1 ¡ t k k²k ² k=1 k=1 Acknowledgement and Z 1 1 ¡ t¡n¡1 Xn 1 N¡lim dt = : The author is grateful to the anonymous referee for a care- ²!0 1 ¡ t k ² k=1 ful checking of the details and for helpful comments that Finally, we have that Ã(¡n) exists and Ã(¡n) = ¡γ + improved this paper.