Journal of Classical Analysis Volume 13, Number 2 (2018), 123–131 doi:10.7153/jca-2018-13-08 ON k–ANALOGUES OF DIGAMMA AND POLYGAMMA FUNCTIONS EMRAH YILDIRIM AND I˙NCI EGE Abstract. In this work, we obtain some integral representations of k-analogue of classical digamma function ψ(x). Then by using the concepts of neutrix and neutrix limit, we generalize the k- ψ ( ) ψ(r)( ) ∈ N digamma function k x and the k-polygamma function k x for all real values of x, r and k > 0. Also further results are given. 1. Introduction The gamma function, which was introduced by Euler, is defined by ∞ − − Γ(x)= tx 1e t dt 0 for x > 0[1]. The logarithmic derivative of gamma function is known as digamma ( or psi) function and it is given by d Γ(x) ψ(x)= logΓ(x)= dx Γ(x) and its some integral representations are defined by 1 − x−1 ∞ −t − ( + )−x ∞ −t −tx ψ( )=−γ + 1 t = e 1 t = e − e x dt dt −t dt (1) 0 1 −t 0 t 0 t 1 − e for positive real values of x,whereγ denotes Euler-Mascheroni constant such that 1 γ = lim 1 + ...+ − logn , n→∞ n in [1]. Mathematics subject classification (2010): 33B15, 33B99. Keywords and phrases: Digamma function, polygamma function, k-digamma function, k-polygamma function, neutrix. This research is supported by Aydın Adnan Menderes University Scientific Research Projects Coordination Unit (BAP) with Project No: FEF-17011. Ð c ,Zagreb 123 Paper JCA-13-08 124 E. YILDIRIM AND I.˙ EGE 1.1. Neutrix calculus In [2], van der Corput introduced the concepts of neutrix and neutrix limit as be- low: DEFINITION 1. (Neutrix) Let N be a nonempty set and let N be a commutative, additive group of functions mapping from N into a commutative, additive group N . The group N is called neutrix if the function which is identically equal to zero is the only constant function occurring N . The function which belongs to N is called “negligible function” in N . Let N be a domain lying in a topological space with a limit point b not belonging to N and N be a commutative additive group of functions defined on N with the following property: “ f ∈ N , lim f (ε)=c (constant) for ε ∈ N then c = 0”. ε→b Then this group N is a neutrix. DEFINITION 2. (Neutrix limit) Let f be a real valued function defined on N and suppose that it is possible to find a constant c such that f (x) − c is negligible in N . Then c is called the neutrix limit of f (x) as x tends to y and denoted by N−lim f (x)=c. (2) x→y In this study, we let N be the neutrix having domain N =(0,∞), range N = R and neglible functions finite linear sums of the functions λ − ε lnr 1 ε, lnr ε (λ < 0, r = 1,2,...) and all functions o(ε) which converge to zero in the normal sense as ε tends to zero. In [9], authors proved that the digamma function can be defined by 1 1 −tx−1 ψ(x)=−γ + N−lim dt (3) ε→0 ε 1 −t for all real values of x. Also they obtained that n 1 ψ(−n)=−γ + ∑ i=1 i for n = 0,1,2,.... In [16], Salem and Kılıc¸man showed the following properties: LEMMA 1. The neutrix limit as ε tends to zero of the integral 1 α− t 1 lnr tdt ε ON k -ANALOGUES OF DIGAMMA AND POLYGAMMA FUNCTIONS 125 exists for all real values of α and r ∈ Z+ and ⎧ (−1)rr! 1 ⎨ α− , α = 0, N−lim t 1 lnr tdt = αr+1 ε→ ε ⎩ 0 0, α = 0 By using Lemma 1, author showed the existence of neutrix limit of the integral part of classical polygamma function for non-positive real values of x and r ∈ Z+ . Then they gave the following definition. ( ) DEFINITION 3. [16] The polygamma function ψ r (x) can be defined by 1 x−1 r ( ) t ln t ψ r (x)=− N−lim dt ε→0 ε 1 −t for all x ∈ R and r ∈ Z+ . We refer the reader to [4, 5, 6, 7, 14, 17, 18, 19] and references therein for more applications of neutrix and neutrix limit to the special functions. 1.2. k-Analogue of classical gamma function and its related function D´ıaz and Pariguan define k-analogue of gamma function by the following Rie- mann integral ∞ k x−1 − t Γk(x)= t e k dt (4) 0 for x ∈ C, Re(x) > 0in[3]. Also, they obtain several results which are generalizations of the classical gamma function. Some of them are given by the followings: Γ ( + )= Γ ( ), k x k x k x (5) x −1 x Γk(x)=k k Γ , (6) k ∞ 1 = − x x γ + x − x . Γ ( ) xk k e k ∏ 1 e nk (7) k x n=1 nk By using the equation (7), Krasniqi in [10] obtain the following series representa- tion of k-digamma function − γ ∞ Ψ ( )=lnk − 1 + x , k x ∑ ( + ) (8) k x i=1 ik x ik for x > 0. The reader can find more properties on k-gamma and its related functions in [10, 11, 12, 13, 15, 20]. 126 E. YILDIRIM AND I.˙ EGE 2. Some integral representations of k-digamma function In this section we seek to redefine k-digamma function as integrals for positive real values of x.Atfirst, we need the following lemma. LEMMA 2. [8] Assume a, b > 0.Then ∞ e−ax − e−bx b dx = ln . 0 x a THEOREM 1. For x > 0, a k-analogue of digamma function can also be ex- pressed by ∞ 1 1 −t − x ψk(x)= e − (1 +tk) k dt. (9) k 0 t Proof. By differentiating the equality (4) with respect to x,wehave ∞ ∞ − − uk 1 − − uk Γ ( )= x 1 k = x 1 k k , k x u e lnudu u e lnu du 0 k 0 andthenbyusingLemma2,weget − − k ∞ k ∞ t − u t Γ ( )=1 x−1 − u e e k x u e k dt du k 0 0 t ∞ ∞ k ∞ 1 1 −t x−1 − u x−1 −uk( 1 +t) = e u e k du− u e k du dt. k 0 t 0 0 The first integral of the integrand on the right side of the last equation is equal to Γk(x). 1 zk For the second one, let us change variable as uk +t = . Then we obtain k k ∞ ∞ x−1 − − k( 1 + ) z − zk dz x 1 u k t = k u e du x−1 e 1 0 0 (1 +tk) k (1 +tk) k ∞ k Γ ( ) 1 x−1 − z k x = x z e k dz = x . (1 +tk) k 0 (1 +tk) k Henceweget ∞ Γk(x) 1 − − x Γ ( )= t − ( + ) k k x e 1 tk dt k 0 t which gives the equation (9). By using previous theorem, we get the following results. ON k-ANALOGUES OF DIGAMMA AND POLYGAMMA FUNCTIONS 127 COROLLARY 1. k-Digamma function can also be given by 1 1− 1 x 1 1 e k t −t k ψk(x)= dt (10) k 0 t(1 −t) for x > 0. REMARK 1. The equation (9) shows that there is a relation between k-digamma and classical digamma functions as lnk 1 x ψ (x)= + ψ (11) k k k k for x > 0. We want to note that the equation (11) can be obtained by taking logarithmic derivative of the equation (6). COROLLARY 2. The k-digamma function can also be defined by ∞ −t − x t ψ ( )=lnk + 1 e − e k k x −t dt (12) k k 0 t 1 − e for x > 0. In this section, we already obtain some integral representations of the k-digamma function by using some properties of integrals and the definition of k-gamma function. In the next theorem, we obtain another representation of it by using the series expression (8). THEOREM 2. The k-digamma function is defined by − γ 1 k−1 − x−1 ψ ( )=lnk + t t k x k dt (13) k 0 1 −t for x > 0. Proof. By using the equation (8), we get − γ ∞ − γ ∞ ∞ ψ ( )=lnk − 1 + 1 − 1 = lnk + 1 − 1 k x ∑ + ∑ ∑ + k x i=1 ik x ik k i=1 ik i=0 x ik ∞ 1 ∞ 1 lnk − γ − + − = + ∑ tik 1dt − ∑ tx ik 1dt k i=1 0 i=0 0 1 ∞ 1 ∞ lnk − γ − + − = + ∑ tik 1dt − ∑ tx ik 1dt k 0 i=1 0 i=0 1 ∞ 1 ∞ lnk − γ − − = + t 1 ∑ tk idt − tx 1 ∑ tk idt. k 0 i=1 0 i=0 128 E. YILDIRIM AND I.˙ EGE Since t ∈ (0,1) and k > 0, both geometric series are convergent. Taking into account that the first series starts with tk and the second with 1, we obtain − γ 1 k 1 − γ 1 k−1 − x−1 ψ ( )=lnk + −1 t − x−1 1 = lnk + t t k x t k dt t k dt k dt k 0 1 −t 0 1 −t k 0 1 −t as desired. Using logarithmic derivative of the equation (5) leads us 1 ψ (x + k)= + ψ (x) (14) k x k for x > 0. We want to note that the formula can also be obtained by using its integral representation (13). 3. Expanding the domain of the k-digamma function The equation (14) helps us to define k-digamma function as 1 ψ (x)=ψ (x + k) − k k x for −k < x < 0.