Higher order generalized geometric polynomials
aa r X i v : . [ m a t h . C A ] D ec Higher order generalized geometric polynomials
Levent Kargın* and Bayram C¸ ekim*** Alanya Alaaddin Keykubat UniversityAkseki Vocational School TR-07630 Antalya TurkeyGazi University, Faculty of Science,Department of Mathematics, Teknikokullar TR-06500,Ankara, [email protected] and **[email protected] 7, 2018
Abstract
According to generalized Mellin derivative [25, Eq. (2.5)], we intro-duce a new family of polynomials called higher order generalized geo-metric polynomials. We obtain some properties of them.We discuss theirconnections to degenerate Bernoulli and Euler polynomials. Furthermore,we find new formulas for the Carlitz’s [9, Eq. (5.4)] and Howard’s [23, Eq.(4.3)] finite sums. Finally, we evaluate several series in closed forms, oneof which has the coefficients include values of the Riemann zeta function.Moreover, we calculate some integrals in terms of generalized geometricpolynomials.
Key words:
Generalized geometric polynomials, Bernoulli polyno-mials, Euler polynomials, Riemann zeta function.
The operator (cid:0) x ddx (cid:1) n , called Mellin derivative [6], has a long mathematical his-tory. As far back as 1740, Euler used the operator as a tool work in his work[19]. The Mellin derivative and its generalizations are used to obtain a newclass of polynomials [5, 17, 18, 25, 32], to evaluate some power series in closedforms [5, 7, 17, 18, 25, 27, 32] and to calculate some integrals [6, 7]. One of thegeneralizations of Mellin derivative is (cid:16) βx − α/β D (cid:17) n h x r/β f ( x ) i = x ( r − nα ) /β n X k =0 S ( n, k ; α, β, r ) β k x k f ( k ) ( x ) , (1)where f is any n -times differentiable function and S ( n, k ; α, β, r ) are generalizedStirling number pair with three free parameters (see Section 2). Stirling numbers1nd their generalizations have many interesting combinatorial interpretations.Besides, these numbers are connected with some well known special polynomialsand numbers [10, 11, 16, 22, 29, 28, 30, 31, 33, 37, 38]. For example, the followinginteresting formulas for Bernoulli numbers B n and Euler polynomials E n ( x )appear in [20]: For all n ≥ B n = n X k =0 ( − k k ! k + 1 (cid:26) nk (cid:27) , E n (0) = n X k =0 ( − k k !2 k (cid:26) nk (cid:27) . (2)From all these motivations, by using the generalized of Mellin derivative in(1), we introduce a new family of polynomials w ( s +1) n ( x ; α, β, r ), called higherorder generalized geometric polynomials, evaluate some power series in closedforms and calculate some integrals. After that, in view of the properties of higherorder generalized geometric polynomials, we derive new explicit formulas fordegenerate Bernoulli polynomials, Bernoulli polynomials and degenerate Eulerpolynomials. As a consequences of these formulas we evaluate Carlitz’s [9, Eq.(5.4)] and Howard’s [23, Eq. (4.3)] sums.The summary by sections is as follows: Section 2 is the preliminary sectionwhere we give definitions and known results needed. In Section 3, we definehigher order generalized geometric polynomials and obtain some properties suchas recurrence relation and generating function of w ( s +1) n ( x ; α, β, r ) . Moreover,we derive new explicit formulas for degenerate Bernoulli polynomials, Bernoullipolynomials and degenerate Euler polynomials. By the help of these formulaswe find new formulas for Carlit’s and Howard’s sums. In final section, we obtainsome integral representation of w ( s +1) n ( x ; α, β, r ). Besides, we evaluate severalpower series in closed forms. For example, in the special cases, the followingseries, where the coefficients include values of the Riemann zeta function, isevaluated in closed form (for any integer n ≥ ∞ X k =2 ζ ( k ) k n k = log 2 + n X k =1 (cid:26) n + 1 k + 1 (cid:27) k ! (cid:0) − − k − (cid:1) ζ ( k + 1) . Throughout this paper we assume that α, β and r are real or complex num-bers. The generalized Stirling numbers of the first kind S ( n, k ; α, β, r ) and of thesecond kind S ( n, k ; α, β, r ) for non-negative integer m and real or complexparameters α, β and r, with ( α, β, r ) = (0 , ,
0) are defined by means of the2enerating function [24] (1 + βt ) α/β − α ! k (1 + βt ) r/β = k ! ∞ X n =0 S ( n, k ; α, β, r ) t n n ! , (1 + αt ) β/α − β ! k (1 + αt ) − r/α = k ! ∞ X n =0 S ( n, k ; α, β, r ) t n n ! , with the convention S ( n, k ; α, β, r ) = S ( n, k ; α, β, r ) = 0 when k > n. As Hsu and Shiue pointed out, the definitions or generating functions gen-eralize various Stirling-type numbers studied previously, such as: i. { S ( n, k ; 0 , , , S ( n, k ; 0 , , } = { s ( n, k ) , S ( n, k ) } = n ( − n − k (cid:2) nk (cid:3) , (cid:8) nk (cid:9)o are the Stirling numbers of both kinds [20, Chapter 6]. ii. { S ( n, k ; α, , − r ) , S ( n, k ; α, , − r ) } = n ( − n − k S ( n, k, r + α | α ) , S ( n, k, r | α ) o are the Howard degenerate weighted Stirling numbers of both kinds [21]. iii. { S ( n, k ; α, , , S ( n, k ; α, , } = n ( − n − k S ( n, k | α ) , S ( n, k | α ) o are the Carlitz’s degenerate Stirling numbers of both kinds [9]. iv. { S ( n, k ; 0 , − , r ) , S ( n, k ; 0 , , r ) } = n ( − n − k (cid:2) n + rk + r (cid:3) r , (cid:8) n + rk + r (cid:9) r o are the r -Stirling numbers of both kinds [8]. v. { S ( n, k ; 0 , β, − , S ( n, k ; 0 , β, } = { w β ( n, k ) , W β ( n, k ) } are the Whitney numbers of the first and second kind [4]. vi. { S ( n, k ; 0 , β, − r ) , S ( n, k ; 0 , β, r ) } = { w β,r ( n, k ) , W β,r ( n, k ) } are the r -Whitney numbers of the first and second kind [31],and so on.According to the generalization of Stirling numbers, Hsu and Shiue [24]defined generalized exponential polynomials S n ( x ) as follows S n ( x ) = n X k =0 S ( n, k ; α, β, r ) x k . (3)Later, Kargin and Corcino [25] gave an equivalent definition for S n ( x ) as S n ( x ) = (cid:2)(cid:0) x nα − r e − x (cid:1)(cid:3) /β (cid:16) βx − α/β D (cid:17) n h ( x r e x ) /β i , (4)3nd using the series e x/β = ∞ X k =0 x k β k k ! , in (4), they obtained the general Dobinksi-type formula [24] e x/β S n ( x ) = ∞ X k =0 ( kβ + r | α ) n x k β k k ! . (5)Here, ( z | α ) n is called the generalized factorial of z with increment α , definedby ( z | α ) n = z ( z − α ) · · · ( z − nα + α ) for n = 1 , , . . . , and ( z | α ) = 1 . Inparticular, we have ( z | n = ( z ) n . Some other properties of S n ( x ) can be found in [14, 15, 24, 36].The generalized geometric polynomials w n ( x ; α, β, r ) are defined by meansof the generalized Mellin derivative as (cid:16) βx − α/β D (cid:17) n (cid:20) x r/β − x (cid:21) = x ( r − nα ) /β − x w n (cid:18) x − x ; α, β, r (cid:19) . These polynomials have the explicit formula w n ( x ; α, β, r ) = n X k =0 S ( n, k ; α, β, r ) β k k ! x k , (6)and the generating function ∞ X n =0 w n ( x ; α, β, r ) t n n ! = (1 + αt ) r/α − x (cid:16) (1 + αt ) β/α − (cid:17) , αβ = 0 . (7)See [25] for details.The higher order degenerate Euler polynomials are defined by means of thegenerating function [9] ∞ X n =0 E ( s ) n ( α ; x ) t n n ! = αt ) /α + 1 ! s (1 + αt ) x/α . (8)From (8), we have lim α → E ( s ) n ( α ; x ) = E ( s ) n ( x ) , where E ( s ) n ( x ) are the higher order Euler polynomials which are defined by thegenerating function ∞ X n =0 E ( s ) n ( x ) t n n ! = (cid:18) e t + 1 (cid:19) s e xt . (9)In special cases, E (1) n ( α ; x ) = E n ( α ; x ) , E (1) n ( x ) = E n ( x ) , E n ( α ; x ) and E n ( x ) are the degenerate Euler and Euler polynomials,respectively.The degenerate Bernoulli polynomials of the second kind are defined bymeans of the generating function [26] ∞ X n =0 B n ( x | α ) t n n ! = α log (1 + αt )(1 + αt ) /α − αt ) x/α . (10)Indeed, we get lim α → B n ( x | α ) = B n ( x ) , where B n ( x ) are the Bernoulli polynomials which are defined by the generatingfunction ∞ X n =0 B n ( x ) t n n ! = te t − e xt , with B n (0) = B n are n -the Bernoulli numbers.Finally, we want to mention Carlitz’s degenerate Bernoulli polynomials de-fined by means of the generating function [9] ∞ X n =0 β n ( α, x ) t n n ! = t (1 + αt ) /α − αt ) x/α , with the relation lim α → β n ( α, x ) = B n ( x )and with β n ( α,
0) = β n ( α ) are n -the degenerate Bernoulli numbers. In this section, the definition of higher order generalized geometric polynomialsand some properties are given. Besides, new explicit formulas for degenerateBernoulli and Euler polynomials are derived. Some special cases of these resultsare studied.For every s ≥ , taking f ( x ) = 1 / (1 − x ) s +1 in (1) and using ∂ k ∂x k f ( x ) = ( s + 1) ( s + 2) . . . ( s + k )(1 − x ) s + k +1 , we have (cid:16) βx − α/β D (cid:17) n " x r/β (1 − x ) s +1 = x ( r − nα ) /β (1 − x ) s +1 n X k =0 S ( n, k ; α, β, r ) (cid:18) s + kk (cid:19) k ! β k (cid:18) x − x (cid:19) k .
5f we define the polynomials w ( s +1) n ( x ; α, β, r ) by w ( s +1) n ( x ; α, β, r ) = n X k =0 S ( n, k ; α, β, r ) (cid:18) s + kk (cid:19) k ! β k x k , (11)we find (cid:16) βx − α/β D (cid:17) n " x r/β (1 − x ) s +1 = x ( r − nα ) /β (1 − x ) s +1 w ( s +1) n (cid:18) x − x ; α, β, r (cid:19) . (12)Note that if ( s, α, β, r ) = (0 , α, β, r ) , we obtain generalized geometric poly-nomials [25], if ( s, α, β, r ) = ( s, , , , we have general geometric polyno-mials [5], if ( s, α, β, r ) = (0 , , , , we obtain geometric polynomials [5], if( s, α, β, r ) = (0 , , β, , we have Tanny-Dowling polynomials [4].For x = 1 and s = 0 in (11) gives the numbers w (1) n (1; α, β, r ) = B n ( α, β, r ) = n X k =0 S ( n, k ; α, β, r ) β k k ! x k , which was defined by Corcino et al. [13]. The combinatorial interpretation andsome other properties can be found in [13, 14]. Moreover, w ( s +1) n ( x ; α, β, r )reduce to barred preferential arrangement numbers r n,s defined by [1, 5] w ( s +1) n (1; 0 , ,
0) = r n,s = n X k =0 (cid:26) nk (cid:27)(cid:18) s + kk (cid:19) k ! , which have interesting combinatorial meaning. Therefore, we may call gener-alized barred preferential arrangement number pair with three free parametersfor B ( s +1) n ( α, β, r ) = n X k =0 S ( n, k ; α, β, r ) (cid:18) s + kk (cid:19) k ! β k x k . The combinatorial interpretation of these numbers may also be studied.On the other hand, we can write w ( s +1) n ( x ; α, β, r ) in the form w ( s +1) n ( x ; α, β, r ) = n X k =0 S ( n, k ; α, β, r ) h s + 1 i k β k x k , since h x i n n ! = (cid:18) x + n − n (cid:19) , where h x i n is the rising factorial defined by h x i n = x ( x + 1) · · · ( x + n − , for n = 1 , , . . . , with h x i = 1 . Furthermore, using the relation h− x i n = ( − n ( x ) n , (13)6f we define w ( − s ) n ( x ; α, β, r ) for every s > w ( − s ) n ( x ; α, β, r ) = n X k =0 S ( n, k ; α, β, r ) ( s ) k ( − β ) k x k , we find (cid:16) βx − α/β D (cid:17) n h x r/β (1 − x ) s i = x ( r − nα ) /β (1 − x ) s w ( − s ) n (cid:18) x − x ; α, β, r (cid:19) (14)by taking f ( x ) = (1 − x ) s in (1).We turn over w ( − s ) n ( x ; α, β, r ) in Section 4.Now, we want to deal with the properties of w ( s ) n ( x ; α, β, r ) . The exponen-tial and higher order generalized geometric polynomials are connected by therelation (for any non-negative integer n and every s > w ( s ) n ( x ; α, β, r ) = 1Γ ( s ) ∞ Z z s − S n ( xβz ) e − z dz, (15)which is verified immediately by using (3). So, we can derive the properties of w ( s ) n ( x ; α, β, r ) from those of S n ( x ). For example, if we use the extension ofSpivey’s Bell number formula to S n ( x ) [25, 36] S n + m ( x ) = n X k =0 m X j =0 (cid:18) nk (cid:19) S ( m, j ; α, β, r ) ( jβ − mα | α ) n − k S k ( x ) x j in (15) and evaluate the the integral, we derive a recurrence relation as w ( s ) n + m ( x ; α, β, r )= n X k =0 m X j =0 (cid:18) nk (cid:19) S ( m, j ; α, β, r ) ( jβ − mα | α ) n − k h s + 1 i j β j w ( s + j ) k ( x ; α, β, r ) x j . As another application of (15), we give the following theorem.
Theorem 1
The exponential generating function for w ( s ) n ( x ; α, β, r ) is ∞ X n =0 w ( s ) n ( x ; α, β, r ) t n n ! = − x (cid:16) (1 + αt ) β/α − (cid:17) s (1 + αt ) r/α , (16) where αβ = 0 . Proof.
From [24, Eq. (12)], let us write the generating function for S n ( x ) inthe form ∞ X n =0 S n ( xβz ) t n n ! = (1 + αt ) r/α exp h xz (cid:16) (1 + αt ) β/α − (cid:17)i . z s − e − z and integrate for z from zero to infinity.In view of (15) this gives ∞ X n =0 w ( s ) n ( x ; α, β, r ) t n n ! = (1 + αt ) r/α Γ ( s ) ∞ Z z s − e − z ( − x ( (1+ αt ) β/α − )) dz. Calculating the integral on the right-hand side completes the proof.Setting x = − w ( s ) n ( − α, β, r ) = w n ( − α, β, r ) = ( r − βs | α ) n . Some other properties of w ( s ) n ( x ; α, β, r ) can be derived from (16).Now, we attend to the connection of w ( s ) n ( x ; α, β, r ) with some degeneratespecial polynomials.If we take x = − / β = 1 and compare with (8), we have w ( s ) n (cid:18) −
12 ; α, , r (cid:19) = E ( s ) n ( α ; r ) . Thus, using (11) yields an explicit formula for higher order degenerate Eulerpolynomials in the following corollary.
Corollary 2
For every s ≥ , we have E ( s +1) n ( α ; r ) = n X k =0 S ( n, k, r | α ) ( − k h s + 1 i k k . (17) When s = 0 , this becomes E n ( α ; r ) = n X k =0 S ( n, k, r | α ) ( − k k !2 k . (18)To the writer’s knowledge, this is a new result. (17) and (18) are new results.From (16) and (9), one can obtainlim α → w ( s ) n (cid:18) −
12 ; α, β, r (cid:19) = E ( s ) n (cid:18) rβ (cid:19) β n . Thus, we have E ( s ) n (cid:18) rβ (cid:19) = n X k =0 W β,r ( n, k ) ( − k h s i k β n − k k , which was given in [33] with a different proof. Moreover, for β = 1 and r = 0the above equation reduce to the right part of (2).Secondly, if we integrate both sides of (7) for x from − , we have ∞ X n =0 t n n ! Z − w n ( x ; α, β, r ) dx = βα log (1 + αt )(1 + αt ) β/α − αt ) r/α . (19)8n view of (10), for β = 1 , the above equation becomes ∞ X n =0 t n n ! Z − w n ( x ; α, , r ) dx = ∞ X n =0 B n ( r | α ) t n n ! . Comparing the coefficients of t n n ! gives Z − w n ( x ; α, , r ) dx = B n ( r | α ) . Finally, using (6) yields the following theorem.
Theorem 3
The following equation holds for degenerate Bernoulli polynomialsof the second kind B n ( r | α ) = n X k =0 S ( n, k, r | α ) ( − k k ! k + 1 . We note that for α → , (19) can be written aslim α → ∞ X n =0 t n n ! Z − w n ( x ; α, β, r ) dx = βte βt − e rt = ∞ X n =0 B n (cid:18) rβ (cid:19) β n t n n ! . Comparing the coefficients of t n n ! in the above equation, we obtainlim α → Z − w n ( x ; α, β, r ) dx = B n (cid:18) rβ (cid:19) β n . Since S ( n, k ; 0 , β, r ) = W β,r ( n, k ) we find B n (cid:18) rβ (cid:19) = n X k =0 W β,r ( n, k ) ( − k k ! β n − k ( k + 1) , which was also given in [33] with a different proof. Moreover, for β = 1 and r = 0 the above equation reduce to the left part of (2).Now, we give a new explicit formula for Carlitz’s degenerate Bernoulli poly-nomials in the following theorem. Theorem 4
For every s ≥ ,β n +1 ( α, r ) − β n +1 ( α, r − s ) = ( n + 1) n X k =0 S ( n, k, r | α ) ( − k h s i k +1 k + 1 . (20)9 roof. If we integrate both sides of (16) for x from − , we have ∞ X n =0 t n n ! Z − w ( s +1) n ( x ; α, β, r ) dx = 1 st " t (1 + αt ) r/α (1 + αt ) β/α − − t (1 + αt ) ( r − βs ) /α (1 + αt ) β/α − . (21)For β = 1, the above equation becomes ∞ X n =0 t n n ! Z − w ( s +1) n ( x ; α, , r ) dx = 1 s " ∞ X n =0 β n +1 ( α, r ) − β n +1 ( α, r − s ) n + 1 t n n ! . Equating the coefficients of t n n ! in the above equation, we obtain Z − w ( s +1) n ( x ; α, , r ) dx = β n +1 ( α, r ) − β n +1 ( α, r − s ) s ( n + 1) . Finally, using (11) yields the desired equation.We note that for s = 1 , (20) becomes β n +1 ( α, r ) − β n +1 ( α, r −
1) = ( n + 1) ( r − | α ) n , which is extension of well-known identity for Bernoulli polynomials B n +1 ( r ) − B n +1 ( r −
1) = ( n + 1) ( r − n . Taking s = r in (20), we have β n +1 ( α, r ) = β n +1 ( α ) + ( n + 1) n X k =0 S ( n, k, r | α ) ( − k h r i k +1 k + 1 . On the other hand, taking s = r in (20) and using the relation [9, Eq. (5.4)] r − X j =0 ( j | α ) m = 1 m + 1 [ β m +1 ( α, r ) − β m +1 ( α )] , we derive the following corollary. Corollary 5
The sums of generalized falling factorials can be expressed as r − X j =0 ( j | α ) n = n X k =0 S ( n, k, r | α ) ( − k h r i k +1 k + 1 , where r is any integer > . s = α in (20) and using the identity [9, Eq. (5.10)] β n ( α, z + α ) = β n ( α, z ) + αnβ n − ( α, z ) , α > , gives the following corollary. Corollary 6
For every α ≥ ,β n ( α, r − α ) = n X k =0 S ( n, k, r | α ) ( − k h α + 1 i k k + 1 . (22)The identity given in [33] can also be derived form (22) for α → . Moreover,when r = 0 , (22) becomes β n ( α, − α ) = n X k =0 S ( n, k | α ) ( − k h α + 1 i k k + 1 . (23)Taking r = α in (22) we have β n ( α ) = n X k =0 S ( n, k, α | α ) ( − k h α + 1 i k k + 1 . (24)Let us return to (21) again. For α → , (21) can be written aslim α → ∞ X n =0 t n n ! Z − w ( s +1) n ( x ; α, β, r ) dx = 1 st (cid:20) te rt e βt − − te ( r − βs ) t e βt − (cid:21) = 1 s ∞ X n =0 B n +1 (cid:16) rβ (cid:17) − B n +1 (cid:16) rβ − s (cid:17) n + 1 β n +1 t n n ! . Since S ( n, k ; 0 , β, r ) = W β,r ( n, k ) , we derive the values of Bernoulli polyno-mials at rational numbers in the following theorem. Theorem 7
For every s ≥ and β = 0 , we have B n +1 (cid:18) rβ (cid:19) − B n +1 (cid:18) rβ − s (cid:19) = ( n + 1) n X k =0 W β,r ( n, k ) ( − k h s i k +1 β n +1 − k ( k + 1) . (25)We note that for β = 1 and r = 0 in (25) we have B n +1 ( − s ) = B n +1 + ( n + 1) n X k =0 (cid:26) nk (cid:27) ( − k +1 h s i k +1 k + 1 . Replacing − s with x in the above equation and using (13) gives the well-knownrelation between Bernoulli polynomials and falling factorial as [35, Eq. (15.39)] B n +1 ( x ) = B n +1 + n X k =0 ( n + 1) k + 1 (cid:26) nk (cid:27) ( x ) k +1 . β = 1 in (25) we have B n +1 ( r ) − B n +1 ( r − s ) = n X k =0 ( − k ( n + 1) k + 1 (cid:26) n + rk + r (cid:27) r h s i k +1 . Thus, taking s = r in the above equation gives a new relation between Bernoullipolynomials and rising factorial in the following corollary. Corollary 8
For every r ≥ , we have B n +1 ( r ) = B n +1 + n X k =0 ( − k ( n + 1) k + 1 (cid:26) n + rk + r (cid:27) r h r i k +1 . For the last consequences of Theorem 7, we deal with the Howard’s identity[23, Eq. (4.3)] m − X j =0 ( r + βj ) n = β n n + 1 (cid:20) B n +1 (cid:18) m + rβ (cid:19) − B n +1 (cid:18) rβ (cid:19)(cid:21) , Replacing − s with m in (25) and using (13) gives that the sums of powers ofintegers can be evaluated as in the following corollary. Corollary 9
Let n and m be non-negative integers with m > and β = 0 . Then, we have m − X j =0 ( r + βj ) n = n X k =0 β k − k + 1 W β,r ( n, k ) ( m ) k +1 . In this section, several examples for the evaluation of some series and integralsare given.For the first example, let us take f ( x ) = cosh ( x/β ) = ∞ X k =0 x k β k (2 k )! , in (1). Then, we have (cid:16) βx − α/β D (cid:17) n h x r/β (cid:16) e x/β + e − x/β (cid:17)i = 2 ∞ X k =0 β k (2 k )! (cid:16) βx − α/β D (cid:17) n h x (2 kβ + r ) /β i = 2 x ( r − nα ) /β ∞ X k =0 (2 kβ + r | α ) n β k (2 k )! x k . (cid:16) βx − α/β D (cid:17) n h x r/β (cid:16) e x/β + e − x/β (cid:17)i = (cid:16) βx − α/β D (cid:17) n h x r/β e x/β i + (cid:16) βx − α/β D (cid:17) n h x r/β e − x/β i = x ( r − nα ) /β e x/β S n ( x ) + x ( r − nα ) /β e − x/β S n ( − x ) . Thus, we have2 ∞ X k =0 (2 kβ + r | α ) n β k (2 k )! x k = e x/β S n ( x ) + e − x/β S n ( − x ) . Moreover, setting x = 2 πiβ in the above equation gives ∞ X k =0 (2 kβ + r | α ) n ( − k (2 π ) k (2 k )! = ⌊ n/ ⌋ X j =1 S ( n, j ; α, β, r ) ( − j (2 πβ ) j . (26)Similarly, taking f ( x ) = sinh x in (1) gives2 ∞ X k =0 ((2 k + 1) β + r | α ) n β k +1 (2 k + 1)! x k +1 = e x/β S n ( x ) − e − x/β S n ( − x )and ∞ X k =0 ((2 k + 1) β + r | α ) n ( − k (2 π ) k (2 k + 1)! = β ⌊ n/ ⌋ X j =1 S ( n, j + 1; α, β, r ) ( − j (2 πβ ) j . (27)Note that (26) and (27) are the generalization of the identities given in [32,Page 403].Now, if we apply (1) to the both sides of the function1(1 − x ) s +1 = ∞ X k =0 (cid:18) s + kk (cid:19) x k and use (12), we obtain (for n, s = 0 , , , . . . ) ∞ X k =0 (cid:18) s + kk (cid:19) ( r + kβ | α ) n x k = 1(1 − x ) s +1 w ( s +1) n (cid:18) x − x ; α, β, r (cid:19) , (28)which is the generalization of [25] ∞ X k =0 ( r + kβ | α ) n x k = 1(1 − x ) w n (cid:18) x − x ; α, β, r (cid:19) . (29)13or the next example, for every s > , if we take f ( x ) = (1 + x ) s = ∞ X k =0 (cid:18) sk (cid:19) x k in (1) and use (14), we derive ∞ X k =0 (cid:18) sk (cid:19) ( r + kβ | α ) n x k = (1 + x ) s w ( − s ) n (cid:18) − x x ; α, β, r (cid:19) , which is the generalization of the identity in [7, Eq. 9].For the last example of the evaluation of the series in closed forms, we dealwith the digamma function ψ ( x ). The digamma function ψ ( x ) can be given bythe Taylor series at x = 1 , [2, Eq. 6.3.14] ψ ( x + 1) = − γ + ∞ X k =1 ζ ( k + 1) ( − k +1 x k , | x | < , (30)where ζ ( s ) is the Riemann zeta function and γ is the Euler’s constant. Taking(30) in (1) gives (cid:16) βx − α/β D (cid:17) n h x r/β ψ ( x + 1) i = x ( r − nα ) /β n X k =0 S ( n, k ; α, β, r ) β k x k ψ ( k ) ( x + 1) , (31)where ψ ( m ) ( x ) is the polygamma function which can be written more compactlyin terms of the Hurwitz zeta function ζ ( s, x ) as [2, Eq. 6.4.10] ψ ( m ) ( x ) = ( − m +1 m ! ζ ( m + 1 , x ) . (32)Here m > , and x is any complex number not equal to a negative integer. Using(30) in the left hand side and (32) in the right hand side of (31), we obtain thefollowing theorem. Theorem 10
For | x | < , ∞ X k =1 ζ ( k + 1) ( r + kβ | α ) n x k (33)= − ( r | α ) n ( ψ (1 − x ) + γ ) + n X k =1 S ( n, k ; α, β, r ) k ! ζ ( k + 1 , − x ) ( βx ) k . As a consequences of Theorem 10, the following sums are obtained:For ( α, β, r ) = (0 , β, r ) , ∞ X k =1 ζ ( k + 1) ( r + kβ ) n x k = − r n ( ψ (1 − x ) + γ )+ n X k =1 W β,r ( n, k ) k ! ζ ( k + 1 , − x ) ( βx ) k . α, β, r ) = (0 , , r ) , ∞ X k =1 ζ ( k + 1) ( r + k ) n x k = − r n ( ψ (1 − x ) + γ )+ n X k =1 (cid:26) n + rk + r (cid:27) r k ! ζ ( k + 1 , − x ) x k . For ( α, β, r ) = (0 , , , using the relation (cid:8) nk (cid:9) = (cid:8) nk (cid:9) = (cid:8) nk (cid:9) , ∞ X k =1 ζ ( k + 1) ( k + 1) n x k = − ( ψ (1 − x ) + γ )+ n X k =1 (cid:26) n + 1 k + 1 (cid:27) k ! ζ ( k + 1 , − x ) x k . (34)Finally, setting x = 1 / ψ (1 /
2) = − γ − ζ ( s, /
2) = (2 s − ζ ( s ) yields ∞ X k =2 ζ ( k ) k n k = log 2 + n X k =1 (cid:26) n + 1 k + 1 (cid:27) k ! (cid:0) − − k − (cid:1) ζ ( k + 1) . Therefore, some special cases of the above equation involving infinite sums ofRiemann zeta function can be listed as ∞ X k =2 ζ ( k )2 k = log 2 , ∞ X k =2 ζ ( k ) k k = log 2 + 34 ζ (2) , ∞ X k =2 ζ ( k ) k k = log 2 + 94 ζ (2) + 148 ζ (3) . Note that similar result of (34) can be found in [7, Proposition 20].Now, we want to prove the equation in (15) by using (1). Applying (1) tothe both sides of the integral1(1 − x ) s = 1Γ ( s ) ∞ Z t s − e − (1 − x ) t dt and using (12) we obtain x ( r − nα ) /β (1 − x ) s +1 w ( s +1) n (cid:18) x − x ; α, β, r (cid:19) = 1Γ ( s ) ∞ Z t s − e − t (cid:16) βx − α/β D (cid:17) n h x r/β e xt i dt.
15t the same time, we have (cid:16) βx − α/β D (cid:17) n h x r/β e xt i = (cid:16) βx − α/β D (cid:17) n ∞ X k =0 t k k ! x k + r/β = ∞ X k =0 t k k ! (cid:16) βx − α/β D (cid:17) n x k + r/β = x ( r − nα ) /β ∞ X k =0 t k k ! ( kβ + r | α ) n x k = x ( r − nα ) /β e xt S n ( xβt ) . Thus, we have a different representation of (15) as1(1 − x ) s +1 w ( s +1) n (cid:18) x − x ; α, β, r (cid:19) = 1Γ ( s ) ∞ Z t s − S n ( xβt ) e − (1 − x ) t dt. Now, we want to add some examples for the evaluation of integrals. Beforegiving the examples we need to mention that in the rest of this section we usethe well-known estimate for the gamma function: | Γ ( x + iy ) | ∼ √ π | y | x − e − x − π | y | , ( | y | → ∞ ) for any fixed real x. This explains the behavior of the gamma functionon vertical lines { t = a + iz : −∞ < z < ∞ , < a < } .For the first example, let us start from the Mellin integral representation[34, Formula 5.37]1(1 + x ) s +1 = 12 πi Γ ( s + 1) a + i ∞ Z a − i ∞ x − t Γ ( t ) Γ ( s + 1 − t ) dt, where s ≥ , < x < . Apply (1) to the both sides of the above integral toobtain (cid:16) βx − α/β D (cid:17) n " x r/β (1 + x ) s +1 = x ( r − nα ) /β πi Γ ( s + 1) a + i ∞ Z a − i ∞ ( r − βt | α ) n x − t Γ ( t ) Γ ( s + 1 − t ) dt. From the left hand side of the above equation, we derive (cid:16) βx − α/β D (cid:17) n " x r/β (1 + x ) s +1 = ∞ X k =0 (cid:18) s + kk (cid:19) ( − k (cid:16) βx − α/β D (cid:17) n x ( kβ + r ) /β = x ( r − nα ) /β ∞ X k =0 (cid:18) s + kk (cid:19) ( r + kβ | α ) n ( − x ) k = x ( r − nα ) /β (1 + x ) s +1 w ( s +1) n (cid:18) − x x ; α, β, r (cid:19) . x ) s +1 w ( s +1) n (cid:18) − x x ; α, β, r (cid:19) = 12 πi Γ ( s + 1) a + i ∞ Z a − i ∞ ( r − βt | α ) n x − t Γ ( t ) Γ ( s + 1 − t ) dt, (35)Secondly, replace x by ( r + kβ ) x , multiply both sides by ( r + kβ | α ) n andsum for k = 0 , , . . . , in the integral: e − x = 12 πi a + i ∞ Z a − i ∞ x − t Γ ( t ) dt. Then, we have e − rx ∞ X k =0 ( r + kβ | α ) n (cid:0) e − βx (cid:1) k = 12 πi a + i ∞ Z a − i ∞ x − t ∞ X k =0 ( kβ + r | α ) n ( kβ + r ) t Γ ( t ) dt. From (29), the above integral becomes e − rx − e − βx w n (cid:18) e − βx − e − βx ; α, β, r (cid:19) = 12 πi a + i ∞ Z a − i ∞ x − t ∞ X k =0 ( kβ + r | α ) n ( kβ + r ) t Γ ( t ) dt. The interesting part of the above integral, given in the following proposition, isappeared when α → . Proposition 11
For all x > , Re ( β ) > , Re ( r ) > n = 0 , , , . . . and a > n + 1 e − rx − e − βx w n (cid:18) e − βx − e − βx ; 0 , β, r (cid:19) = β n πi a + i ∞ Z a − i ∞ ( βx ) − t ζ (cid:18) t − n, rβ (cid:19) Γ ( t ) dt. (36)We note that for α → , generalized geometric polynomials become w n ( x ; 0 , β, r ) = n X k =0 W β,r ( n, k ) β k k ! x k . Moreover, for β = 1 and n = 0 in (36), we have the well-known inverseMellin transformation of Hurwitz zeta function [3, Theorem 12.2]. Moreover,(35) and (36) are the generalization of identities in [7, Proposition 16]. References [1] C. Ahlbach, J. Usatine, N. Pippenger, Barred preferential arrangement,Electron. J. Combin. 20 (2) (2013) r -Stirling numbers, Discrete Math. 49 (1984) 241–259.[9] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Util.Math.15 (1979) 51–88.[10] M. Cenkci, F. T. Howard, Notes on degenerate numbers, Discrete Math307 (2007) 2359-2375.[11] M. Cenkci, An explicit formula for generalized potential polynomials andits applications, Discrete Math, 309 (2009) 1498-1510.[12] L. Comtet, Advanced Combinatorics, D. Reidel Dordrecht 1974.[13] R. B. Corcino, Some theorems on generalized Stirling numbers, Ars Com-bin. 60 (2001) 273–286.[14] R. B. Corcino, C. B. Corcino, On generalized Bell polynomials, DiscretDyn Nat Soc. 2011 2011 21p.[15] R. B. Corcino, M. B. Montero, C. B. Corcino, On generalized Bell numbersfor complex argument, Util.Math. 88 (2012) 267-279.[16] M.C. Da˘glı, M. Can, On reciprocity formula of character Dedekind sumsand the integral of products of Bernoulli polynomials, J. Number Theory (2015) 105–124.[17] A. Dil, V. Kurt, Polynomials related to harmonic numbers and evaluationof harmonic number series I, Integers 12 (2012), A38.[18] A. Dil, V. Kurt, Polynomials related to harmonic numbers and evaluationof harmonic number series II, Appl. Anal. Discrete Math. 5 (2011), 212–229.[19] L. Euler, Institutiones Calculi Differentialis, Vol. II, Academie ImperialisScientiarum Petropolitanae, 1755.1820] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics,Addison-Wesley Publ. Co. New York 1994.[21] F.T.Howard, Degenerate weighted Stirling numbers, Discrete Math. 57(1985) 45–58.[22] F.T. Howard, Explicit formulas for degenerate Bernoulli numbers, DiscreteMath. 162 (1996) 175–185.[23] F. T. Howard, Sums of powers of integers via generating functions, Fi-bonacci Quart. 34 (3) (1996) 244–256.[24] L. C. Hsu, P. J. S. Shiue, A unified approach to generalized Stirling num-bers, Adv. in Appl. Math., 20 3 (1998) 366–384.[25] L. Kargin, R. B. Corcino, Generalization of Mellin derivative and its appli-cations, Integral Transforms Spec. Funct. 27 (8) (2016) 620–631.[26] T. Kim, Jong Jin Seo, Degenerate Bernoulli Numbers and Polynomials ofthe Second Kind, International Journal of Mathematical Analysis 9 (26)2015 1269-1278.[27] P. M. Knopf, The operator (cid:0) x ddx (cid:1) n and its application to series, Math. Mag.76 (5) (2003) 364–371.[28] M. Merca, A new connection between r -Whitney numbers and Bernoullipolynomials. Integral Transforms Spec. Funct. 25 (12) (2014) 937–942.[29] M. Merca, A connection between Jacobi–Stirling numbers and Bernoullipolynomials, J. Number Theory