EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6202 ISSN 1307-5543 – ejpam.com Published by New York Business Global The r-Stirling Genocchi Numbers Roberto B. Corcino1,2,∗, Vernard P. Dechosa2 1 Research Institute for Computational Mathematics and Physics, Cebu Normal University, 6000 Cebu City, Philippines 2 Mathematics Department, Cebu Normal University, 6000 Cebu City, Philippines Abstract. This paper introduces the r-Stirling Genocchi numbers, a new sequence derived by combining Broder’s r-Stirling numbers with the classical Genocchi numbers, which are closely related to Bernoulli numbers and have notable applications in algebraic combinatorics. While the original r-Stirling numbers were developed using combinatorial methods to count set partitions, this study adopts an algebraic approach to explore the properties of the new sequence. Through tools like generating functions, recurrence relations, and algebraic transformations, the paper uncovers deeper structural insights and highlights the broader mathematical connections between partition theory, number theory, and combinatorial analysis. 2020 Mathematics Subject Classifications: 11B68, 11B73, 05A15 Key Words and Phrases: Genocchi numbers, r-Stirling numbers, recurrence relations, gener- ating function, Bernoulli numbers 1. Introduction Stirling numbers were first introduced by James Stirling with pairs of numbers in his book Methodus differentialis usually denoted by s(n, k) and S(n, k) where s(n, k) referred to as Stirling numbers of the first kind and S(n, k) is referred to as Stirling numbers of the second kind. [1] Over the years, many mathematics enthusiasts have investigated and expanded these two special numbers. One of them is Broder [2], where he developed both types of r- Stirling numbers and imposed a constraint on the first r elements, requiring them to be arranged in different cycles or partition in different subsets. The r-Stirling numbers are as follows: [ n k ] r ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6202 Email addresses: rcorcino@yahoo.com (R. B. Corcino), dechosav@cnu.edu.ph (V. P. Dechosa) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) 2 of 26 can be represented as the number of permutations of the set {1, ..., n} having m cycles, such that the numbers 1, 2, ..., r are in distinct.{ n k } r can be represented as the number of partitions of the set {1, ..., n} into m non-empty disjoint subsets such that the numbers 1, 2, ..., r are in distinct subsets. Since Broder’s formulation was primarily combinatorial, Corcino et.al [3] later intro- duced an algebraic approach by defining r-Stirling numbers using exponential generating functions: ∞∑ n=0 ̂[n+ r k + r ] r tn n! = ( 1 1 + t )r lnk(1 + t) k! ∞∑ n=0 { n+ r k + r } r tn n! = ert(et − 1)k k! A natural extension of these ideas leads to Genocchi numbers, first studied by Angelo Genocchi. The Genocchi numbers Gn satisfy the generating function given as follows ∞∑ n=0 Gn tn n! = 2t et + 1 Moreover, the Genocchi polynomials and Genocchi polynomials of higher order, which are respectively defined by ∞∑ n=0 Gn(x) tn n! = 2t et + 1 ext, |t| < π, (1.1) ∞∑ n=0 G(k) n (x) tn n! = ( 2t et + 1 )k ext, (1.2) (See [4–6]). It is important to note that the Genocchi polynomials satisfy the following relation Gn(x) = n∑ m=0 ( n m ) Gn−mxm (1.3) expressing Gn(x) as polynomial in x. In relation to this, we define Genocchi factorial polynomials, denoted by Gn(x), as follows Gn(x) = n∑ m=0 ( n m ) Gn−m(x)m. (1.4) 3 of 26 Different variations of Genocchi numbers and polynomials and their properties are dis- cussed in [7–9] In this paper, it presents a new class of numbers called r-Stirling Genocchi numbers, created by merging the r-Stirling numbers with Genocchi numbers and explore its behav- ior by generating results.The r-Stirling Genocchi numbers defined by means of exponential generating function are as follows: ∞∑ n=0 SG1 n(k; r) tn n! = ( 1 1 + t )r 2t lnk(1 + t) k!(et + 1) (1) ∞∑ n=0 SG2 n(k; r) tn n! = 2tert(et − 1)k k!(et + 1) (2) where SG1 n(k; r) referred to as the r-Stirling Genocchi number of the First Kind and SG2 n(k; r) is the r-Stirling Genocchi number of the Second Kind. Theorem 1.1. Convolution Formula of the r-Stirling Numbers and Genocchi Numbers are given as follows: SG1 n(k; r) = n∑ j=k ̂[j + r k + r ] r ( n j ) Gn−j (3) SG2 n(k; r) = n∑ j=k { j + r k + r } r ( n j ) Gn−j (4) where n ≥ k otherwise SG1 n(k; r) = SG2 n(k; r) = 0 Proof. The Exponential Generating Function in (1) can be written as ∞∑ n=0 SG1 n(k; r) tn n! = ( 1 1 + t )r 2t lnk(1 + t) k!(et + 1) = ( 1 1 + t )r lnk(1 + t) k! 2t (et + 1) = ( ∞∑ n=0 ̂[n+ r k + r ] r tn n! )( ∞∑ n=0 Gn tn n! ) = ∞∑ n=0 n∑ j=0 ̂[j + r k + r ] r tj j! Gn−j tn−j (n− j)! = ∞∑ n=0  n∑ j=0 ̂[j + r k + r ] r n! (n− j)!j! Gn−j  tn n! 4 of 26 = ∞∑ n=0  n∑ j=0 ̂[j + r k + r ] r ( n j ) Gn−j  tn n! Comparing the coefficients of tn n! yields the desired convolution formula in (3). On the other hand, the exponential generating function in (2) can be written as ∞∑ n=0 SG2 n(k; r) tn n! = 2tert(et − 1)k k!(et + 1) = ert(et − 1)k k! 2t (et + 1) = ( ∞∑ n=0 { n+ r k + r } r tn n! )( ∞∑ n=0 Gn tn n! ) = ∞∑ n=0 n∑ j=0 { j + r k + r } r tj j! Gn−j tn−j (n− j)! = ∞∑ n=0  n∑ j=0 { j + r k + r } r n! (n− j)!j! Gn−j  tn n! = ∞∑ n=0  n∑ j=0 { j + r k + r } r ( n j ) Gn−j  tn n! Comparing the coefficients of tn n! yields the desired convolution formula in (4). Theorem 1.2. The Horizontal Generating Function for both kinds of r-Stirling Genocchi Numbers are given as follows: Gn(z − r) = n∑ k=0 SG1 n(k; r)z k (5) Gn(z + r) = n∑ k=0 SG2 n(k; r)z k (6) Proof. The exponential generating function of (1) can be written as ∞∑ k=0 { ∞∑ n=k SG1 n(k; r) tn n! } zk = ( 1 1 + t )r 2t (et + 1) ∑ k≥0 lnk(1 + t) k! zk = ( 1 1 + t )r 2t (et + 1) ∑ k≥0 [z ln(1 + t)]k k! 5 of 26 = ( 1 1 + t )r eln(1+t)z 2t (et + 1) = ∑ n≥0 ( z − r n ) tn ∞∑ n=0 Gn tn n! = ∑ n≥0 (z − r)n tn n! ( ∞∑ n=0 Gn tn n! ) = ∞∑ n=0 n∑ m=0 (z − r)m tm m! Gn−m tn−m (n−m)! = ∞∑ n=0 { n∑ m=0 (z − r)m 1 (n−m)!m! Gn−m } tn = ∞∑ n=0 { n∑ m=0 (z − r)m ( n m ) Gn−m } tn n! = ∞∑ n=0 { n∑ m=0 ( n m ) Gn−m(z − r)m } tn n! Rewriting ∞∑ n=0 { n∑ k=0 SG1 n(k; r)z k } tn n! = ∞∑ n=0 { n∑ m=0 ( n m ) Gn−m(z − r)m } tn n! Comparing the coefficients of tn n! , we have n∑ k=0 SG1 n(k; r)z k = n∑ m=0 ( n m ) Gn−m(z − r)m = Gn(z − r) Similarly (2) can be written as ∞∑ k=0 { ∞∑ n=k SG2 n(k; r) tn n! } zk = ∞∑ k=0 { 2tert(et − 1)k k!(et + 1) } zk = 2tert (et + 1) ∞∑ k=0 ( z k ) (et − 1)k = 2tert (et + 1) (1 + (et − 1))z = 2t (et + 1) ert(1 + (et − 1))z 6 of 26 = e(z+r)t 2t (et + 1) = ( ∞∑ n=0 (z + r)n tn n! )( ∞∑ n=0 Gn tn n! ) = ∞∑ n=0 n∑ m=0 (z + r)m tm m! Gn−m tn−m (n−m)! = ∞∑ n=0 { n∑ m=0 (z + r)m 1 (n−m)!m! Gn−m } tn = ∞∑ n=0 { n∑ m=0 (z + r)m ( n m ) Gn−m } tn n! = ∞∑ n=0 { n∑ m=0 ( n m ) Gn−m(z + r)m } tn n! Rewriting ∞∑ n=0 { n∑ k=0 SG2 n(k; r)z k } tn n! = ∞∑ n=0 { n∑ m=0 ( n m ) Gn−m(z + r)m } tn n! Comparing the coefficients of tn n! , we have n∑ k=0 SG2 n(k; r)z k = n∑ m=0 ( n m ) Gn−m(z + r)m = Gn(z + r) Hence, we proved the horizontal generating function in (5) and (6). An essential characteristic of a special number is its explicit formula, which is valuable for directly calculating the number’s value for particular parameter inputs. The following theorem presents the explicit formula for the r-Stirling Genocchi numbers. Theorem 1.3. The formula for the first kind of r-Stirling Genocchi Numbers is explicitly stated as follows: SG1 n(k; r) = n∑ m=0 n∑ j=m m−k∑ f=0 f∑ d=i (−1)n−j+d+fGj−m ( j m )( n j )( f d ) ( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! rn−j 7 of 26 Proof. The exponential generating function in (1) composed of three functions. The first function can be expressed as( 1 1− t )r = (1− t)−r = ∑ n≥0 ( −r n ) tn = ( −r 0 ) (−t)0 + ∑ n>0 n ( −r n ) tn By Newtons Binomial Theorem,( 1 1− t )r = ∑ n≥0 (−r)(−r − 1) . . . (−r − n+ 1) n! (−t)n = ∑ n≥0 (−1)n (r)(r + 1) . . . (r + n− 1) n! (−1)n(t)n And by the definition of a rising factorial,( 1 1− t )r = ∑ n≥0 (−1)nrn tn n! where rn = r(r + 1) . . . (r + n− 1) The second function can be expresses as the Genocchi number, 2t (et + 1) = ∞∑ n=0 Gn tn n! Lastly the third function can be expressed 1 k! [ln 1 + t]k = ∑ n≥k s(n, k) tn n! Hence, using Cauchy’s Rule for the product of power series, we have n∑ k≥0 SG1 n(k; r) tn n! = ∑ n≥0 (−1)nrn tn n! ( ∞∑ n=0 Gn tn n! )∑ n≥k s(n, k) tn n!  8 of 26 = ∑ n≥0 (−1)nrn tn n! ( ∞∑ n=0 { n∑ m=k s(m, k) ( n m ) Gn−m } tn n! ) = ∞∑ n=0 n∑ j=0 j∑ m=k s(m, k) ( j m ) Gj−m tj j! (−1)n−jrn−j tn−j (n− j)! = ∞∑ n=0 n∑ j=0 j∑ m=k s(m, k) ( j m ) Gj−m tj−j+n (n− j)!j! (−1)n−jrn−j = ∞∑ n=0  n∑ m=0 n∑ j=m s(m, k) ( j m ) Gj−m 1 (n− j)!j! (−1)n−jrn−j  tn = ∞∑ n=0  n∑ m=0 n∑ j=m s(m, k) ( j m ) Gj−m ( n j ) (−1)n−jrn−j  tn n! Comparing the coefficients of tn n! , SG1 n(k; r) = n∑ m=0 n∑ j=m s(m, k) ( j m ) Gj−m ( n j ) (−1)n−jrn−j Using the Schlömilch Formula for the Stirling numbers of the first kind, s(n, k) = n−k∑ r=0 r∑ j=i (−1)j+r ( r j )( n− 1 + r n− k + r )( 2n− k n− k + r ) (r − j)n−k+r r! Then the Schlömilch Formula for the r-Stirling Genocchi number of the first kind is given by SG1 n(k; r) = n∑ m=0 n∑ j=m ( j m ) Gj−m ( n j ) (−1)n−jrn−j m−k∑ f=0 f∑ d=i (−1)d+f ( f d )( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! = n∑ m=0 n∑ j=m m−k∑ f=0 f∑ d=i ( j m ) Gj−m ( n j ) (−1)n−jrn−j (−1)d+f ( f d )( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! 9 of 26 = n∑ m=0 n∑ j=m m−k∑ f=0 f∑ d=i (−1)n−j+d+fGj−m( j m )( n j )( f d )( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! rn−j Theorem 1.4. The formula for the second kind of r-Stirling Genocchi Numbers is explic- itly defined by the following formula: SG2 n(k; r) = n∑ m=0 { m+ r k + r } r ( n m ) Gn−m Proof. The exponential generating function in (2) can be written as n∑ n≥k k!SG2 n(k; r) tn n! = 2tert (et + 1 k∑ i=0 ( k i ) (et)k−i(−1)i = 2t (et + 1) k∑ i=0 ( k i ) ert(et)k−i(−1)i = 2t (et + 1) k∑ i=0 ( k i ) ert+(k−i)t(−1)i = k∑ i=0 ( k i ) ert(et)k−i(−1)i ∞∑ n=0 Gn tn n! =  k∑ i=0 ( k i )∑ n≥0 [((k − i) + r)t]n n! (−1)i ( ∞∑ n=0 Gn tn n! ) =  ∞∑ n≥0 { k∑ i=0 (−1)i ( k i ) ((k − i) + r)n } tn n! ( ∞∑ n=0 Gn tn n! ) =  ∞∑ n≥0 k! { n+ r k + r } r tn n! ( ∞∑ n=0 Gn tn n! ) = ∞∑ n=0 n∑ m=0 k! { m+ r k + r } r tm m! Gn−m tn−m (n−m)! = ∞∑ n=0 { n∑ m=0 k! { m+ r k + r } r 1 (n−m)!m! Gn−m } tn 10 of 26 = ∞∑ n=0 { n∑ m=0 k! { m+ r k + r } r ( n m ) Gn−m } tn n! Comparing the coefficients of tn n! , we have k!SG2 n(k; r) = n∑ m=0 k! { m+ r k + r } r ( n m ) Gn−m SG2 n(k; r) = n∑ m=0 { m+ r k + r } r ( n m ) Gn−m The following sections investigates the convolutions of r-Stirling numbers with the Genocchi polynomials and higher order genocchi polynomials. 2. r-Stirling Genocchi Polynomials The Genocchi Polynomials satisfy the relation ∞∑ n=0 Gn(x) tn n! = 2t et + 1 ext The r-Stirling Genocchi Polynomials defined by means of exponential generating function are as follows: ∞∑ n=0 SG1 n(x; k; r) tn n! = ( 1 1 + t )r 2text lnk(1 + t) k!(et + 1) (7) ∞∑ n=0 SG2 n(x; k; r) tn n! = 2textert(et − 1)k k!(et + 1) (8) Theorem 2.1. Convolution Formula of the r-Stirling Numbers and Genocchi Polynomials are given as follows: SG1 n(x; k, r) = n∑ i=0 n∑ j=i ̂[ n− j + r k + r ] r ( n j )( j i ) Gj−ix i (9) SG2 n(x; k, r) = n∑ i=0 n∑ j=i { n− j + r k + r } r ( n j )( j i ) Gj−ix i (10) where n ≥ k otherwise SG1 n(k; r) = SG2 n(k; r) = 0 11 of 26 Proof. The Exponential Generating Function in (7) can be written as ∞∑ n=0 SG1 n(x; k, r) tn n! = ( 1 1 + t )r 2text lnk(1 + t) k!(et + 1) = ( 1 1 + t )r lnk(1 + t) k! 2text (et + 1) = ( ∞∑ n=0 ̂[n+ r k + r ] r tn n! )( ∞∑ n=0 Gn(x) tn n! ) = ∞∑ n=0 n∑ j=0 ̂[j + r k + r ] r tj j! Gn−j(x) tn−j (n− j)! = ∞∑ n=0 n∑ j=0 ̂[ n− j + r k + r ] r tn−j (n− j)! Gj(x) tj j! Since Gj(x) = ∑j i=0 ( j i ) Gix j−i, then it follows ∞∑ n=0 SG1 n(x; k, r) tn n! = ∞∑ n=0 n∑ j=0 ( n j ) ̂[ n− j + r k + r ] r j∑ i=0 ( j i ) Gj−ix i t n n! = ∞∑ n=0  n∑ i=0 n∑ j=i ̂[ n− j + r k + r ] r ( n j )( j i ) Gj−ix i  tn n! Comparing the coefficients of tn n! yields the desired convolution formula in (9). On the other hand, the exponential generating function in (8) can be written as ∞∑ n=0 SG2 n(x; k, r) tn n! = 2textert(et − 1)k k!(et + 1) = ert(et − 1)k k! 2text (et + 1) = ( ∞∑ n=0 { n+ r k + r } r tn n! )( ∞∑ n=0 Gn(x) tn n! ) = ∞∑ n=0 n∑ j=0 { j + r k + r } r tj j! Gn−j(x) tn−j (n− j)! = ∞∑ n=0 n∑ j=0 { n− j + r k + r } r tn−j (n− j)! Gj(x) tj j! = ∞∑ n=0 n∑ j=0 { n− j + r k + r } r tn−j (n− j)! j∑ i=0 ( j i ) Gix j−i t j j! 12 of 26 = ∞∑ n=0  n∑ i=0 n∑ j=i { n− j + r k + r } r ( n j )( j i ) Gj−ix i  tn n! Comparing the coefficients of tn n! yields the desired convolution formula in (10). Theorem 2.2. The Horizontal Generating Function for both kinds of r-Stirling Genocchi Polynomials are given as follows: Gn(x+ z − r) = n∑ k=0 SG1 n(x; k, r)z k (11) Gn(x+ z + r) = n∑ k=0 SG2 n(x; k, r)z k (12) Proof. The exponential generating function of (7) can be written as ∞∑ k=0 { ∞∑ n=k SG1 n(x; k, r) tn n! } zk = ( 1 1 + t )r 2text (et + 1) ∑ k≥0 lnk(1 + t) k! zk = ( 1 1 + t )r 2text (et + 1) ∑ k≥0 [z ln(1 + t)]k k! = ( 1 1 + t )r eln(1+t)z 2text (et + 1) = ∑ n≥0 ( z − r n ) tn ∞∑ n=0 Gn(x) tn n! = ∑ n≥0 (z − r)n tn n! ( ∞∑ n=0 Gn(x) tn n! ) = ∞∑ n=0 n∑ m=0 (z − r)m tm m! Gn−m(x) tn−m (n−m)! = ∞∑ n=0 { n∑ m=0 (z − r)m 1 (n−m)!m! Gn−m(x) } tn = ∞∑ n=0 { n∑ m=0 (z − r)m ( n m ) Gn−m(x) } tn n! = ∞∑ n=0 { n∑ m=0 ( n m ) Gn−m(x)(z − r)m } tn n! Rewriting 13 of 26 ∞∑ n=0 { n∑ k=0 SG1 n(x; k, r)z k } tn n! = ∞∑ n=0 { n∑ m=0 ( n m ) Gn−m(x)(z − r)m } tn n! Comparing the coefficients of tn n! , we have n∑ k=0 SG1 n(x; k, r)z k = n∑ m=0 ( n m ) Gn−m(x)(z − r)m Applying the Addition Formula, we have n∑ k=0 SG1 n(x; k, r)z k = Gn(x+ z − r) Similarly (8) can be written as ∞∑ k=0 { ∞∑ n=k SG2 n(x; k, r) tn n! } zk = ∞∑ k=0 { 2textert(et − 1)k k!(et + 1) } zk = 2textert (et + 1) ∞∑ k=0 ( z k ) (et − 1)k = 2textert (et + 1) (1 + (et − 1))z = 2text (et + 1) ert(1 + (et − 1))z = e(z+r)t 2text (et + 1) = ( ∞∑ n=0 (z + r)n tn n! )( ∞∑ n=0 Gn(x) tn n! ) = ∞∑ n=0 n∑ m=0 (z + r)m tm m! Gn−m(x) tn−m (n−m)! = ∞∑ n=0 { n∑ m=0 (z + r)m 1 (n−m)!m! Gn−m(x) } tn = ∞∑ n=0 { n∑ m=0 (z + r)m ( n m ) Gn−m(x) } tn n! = ∞∑ n=0 { n∑ m=0 ( n m ) Gn−m(x)(z + r)m } tn n! 14 of 26 Rewriting ∞∑ n=0 { n∑ k=0 SG2 n(x; k, r)z k } tn n! = ∞∑ n=0 { n∑ m=0 ( n m ) Gn−m(x)(z + r)m } tn n! Comparing the coefficients of tn n! , we have n∑ k=0 SG2 n(x; k, r)z k = n∑ m=0 ( n m ) Gn−m(x)(z + r)m Applying the Addition Formula, we have n∑ k=0 SG2 n(x; k, r)z k = Gn(x+ z + r) Hence, we proved the horizontal generating function in (11) and (12). Theorem 2.3. The formula for the first kind of r-Stirling Genocchi Polynomials is ex- plicitly stated as follows: SG1 n(x; k, r) = n∑ m=0 n∑ j=m j−m∑ i=0 m−k∑ f=0 f∑ d=i (−1)n−j+d+fGj−m−ix i ( j m )( j −m i )( n j )( f d ) ( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! rn−j Proof The exponential generating function in (7) composed of three functions. The first function can be expressed as( 1 1− t )r = ∑ n≥0 (−1)nrn tn n! where rn = r(r + 1) . . . (r + n− 1) The second function can be expresses as the Genocchi Polynomial, 2t (et + 1) ext = ∞∑ n=0 Gn(x) tn n! Lastly the third function can be expressed 1 k! [ln (1 + t)]k = ∑ n≥k s(n, k) tn n! Hence, using Cauchy’s Rule for the product of power series, we have n∑ k≥0 SG1 n(x; k, r) tn n! = ∑ n≥0 (−1)nrn tn n! ( ∞∑ n=0 Gn(x) tn n! )∑ n≥k s(n, k) tn n!  15 of 26 = ∑ n≥0 (−1)nrn tn n! ( ∞∑ n=0 { n∑ m=k s(m, k) ( n m ) Gn−m(x) } tn n! ) = ∞∑ n=0 n∑ j=0 j∑ m=k s(m, k) ( j m ) Gj−m(x) tj j! (−1)n−jrn−j tn−j (n− j)! = ∞∑ n=0 n∑ j=0 j∑ m=k s(m, k) ( j m ) Gj−m(x) tj−j+n (n− j)!j! (−1)n−jrn−j = ∞∑ n=0  n∑ m=0 n∑ j=m s(m, k) ( j m ) Gj−m(x) 1 (n− j)!j! (−1)n−jrn−j  tn = ∞∑ n=0  n∑ m=0 n∑ j=m s(m, k) ( j m ) j−m∑ i=0 ( j −m i ) Gj−m−ix i ( n j ) (−1)n−jrn−j  tn n! = ∞∑ n=0  n∑ m=0 n∑ j=m j−m∑ i=0 s(m, k) ( j m )( j −m i ) Gj−m−ix i ( n j ) (−1)n−jrn−j  tn n! Comparing the coefficients of tn n! , SG1 n(x; k, r) = n∑ m=0 n∑ j=m m∑ i=0 ( j m )( m i )( n j ) Gm−ix is(m, k)(−1)n−jrn−j Using the Schlömilch Formula for the Stirling numbers of the first kind, s(n, k) = n−k∑ r=0 r∑ j=i (−1)j+r ( r j )( n− 1 + r n− k + r )( 2n− k n− k + r ) (r − j)n−k+r r! Then the Schlömilch Formula for the r-Stirling Genocchi Polynomial of the first kind is given by SG1 n(x; k, r) = n∑ m=0 n∑ j=m j−m∑ i=0 ( j m )( j −m i )( n j ) Gj−m−ix i(−1)n−jrn−j m−k∑ f=0 f∑ d=i (−1)d+f ( f d )( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! = n∑ m=0 n∑ j=m j−m∑ i=0 m−k∑ f=0 f∑ d=i ( j −m i )( j m ) Gj−m−ix i ( n j ) (−1)n−jrn−j (−1)d+f ( f d )( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! 16 of 26 = n∑ m=0 n∑ j=m j−m∑ i=0 m−k∑ f=0 f∑ d=i (−1)n−j+d+fGj−m−ix i ( j m )( j −m i )( n j )( f d )( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! rn−j Theorem 2.4. The formula for the second kind of r-Stirling Genocchi Polynomials is explicitly defined by the following formula: SG2 n(x; k, r) = n∑ m=0 j−m∑ i=0 { m+ r k + r } r ( n m )( j −m i ) Gm−ix i Proof The exponential generating function in (8) can be written as n∑ n≥k k!SG2 n(x; k, r) tn n! = 2textert (et + 1) k∑ i=0 ( k i ) (et)k−i(−1)i = 2text (et + 1) k∑ i=0 ( k i ) ert(et)k−i(−1)i = 2text (et + 1) k∑ i=0 ( k i ) ert+(k−i)t(−1)i = k∑ i=0 ( k i ) ert(et)k−i(−1)i ∞∑ n=0 Gn(x) tn n! =  k∑ i=0 ( k i )∑ n≥0 [((k − i) + r)t]n n! (−1)i ( ∞∑ n=0 Gn(x) tn n! ) =  ∞∑ n≥0 { k∑ i=0 (−1)i ( k i ) ((k − i) + r)n } tn n! ( ∞∑ n=0 Gn(x) tn n! ) =  ∞∑ n≥0 k! { n+ r k + r } r tn n! ( ∞∑ n=0 Gn(x) tn n! ) = ∞∑ n=0 n∑ m=0 k! { m+ r k + r } r tm m! Gn−m(x) tn−m (n−m)! = ∞∑ n=0 { n∑ m=0 k! { m+ r k + r } r 1 (n−m)!m! Gn−m(x) } tn = ∞∑ n=0 { n∑ m=0 k! { m+ r k + r } r ( n m ) j−m∑ i=0 ( j −m i ) Gm−ix i } tn n! 17 of 26 = ∞∑ n=0 { n∑ m=0 j−m∑ i=0 k! { m+ r k + r } r ( n m )( j −m i ) Gm−ix i } tn n! Comparing the coefficients of tn n! , we have k!SG2 n(x; k, r) = n∑ m=0 j−m∑ i=0 k! { m+ r k + r } r ( n m )( j −m i ) Gm−ix i SG2 n(x; k, r) = n∑ m=0 j−m∑ i=0 { m+ r k + r } r ( n m )( j −m i ) Gm−ix i 3. r-Stirling Genocchi Polynomials of Higher Order The Genocchi Polynomials of Higher Order satisfy the relation ∞∑ n=0 Gw n (x) tn n! = ( 2t et + 1 )w ext, (see [4]). Now, we define the r-Stirling Genocchi Polynomials of higher order by means of exponential generating function are as follows: ∞∑ n=0 SG1w n (x; k, r) tn n! = ( 1 1 + t )r ( 2t et + 1 )w ext lnk(1 + t) k! (13) ∞∑ n=0 SG2w n (x; k, r) tn n! = ( 2t et + 1 )w extert(et − 1)k k!(et + 1) (14) We can also establish some properties parallel to those of r-Stirling Genocchi polynomials. Theorem 3.1. Convolution Formula of the r-Stirling Numbers and Genocchi Polynomials of higher order are given as follows: SG1w n (x; k, r) = n∑ j=k ̂[j + r k + r ] r ( n j ) Gw n−j(x) (15) SG2w n (x; k, r) = n∑ j=k { j + r k + r } r ( n j ) Gw n−j(x) (16) where n ≥ k otherwise SG1w n (x; k, r) = SG2w n (x; k, r) = 0 Proof. The Exponential Generating Function in (13) can be written as ∞∑ n=0 SG1w n (x; k, r) tn n! = ( 1 1 + t )r ( 2t et + 1 )w ext lnk(1 + t) k! 18 of 26 = ( 1 1 + t )r lnk(1 + t) k! ( 2t et + 1 )w ext = ( ∞∑ n=0 ̂[n+ r k + r ] r tn n! )( ∞∑ n=0 Gw n (x) tn n! ) = ∞∑ n=0 n∑ j=0 ̂[j + r k + r ] r tj j! Gw n−j(x) tn−j (n− j)! = ∞∑ n=0  n∑ j=0 ̂[j + r k + r ] r n! (n− j)!j! Gw n−j(x)  tn n! = ∞∑ n=0  n∑ j=0 ̂[j + r k + r ] r ( n j ) Gw n−j(x)  tn n! Comparing the coefficients of tn n! yields the desired convolution formula in (15). On the other hand, the exponential generating function in (14) can be written as ∞∑ n=0 SG2w n (x; k, r) tn n! = ( 2t et + 1 )w extert(et − 1)k k!(et + 1) = ert(et − 1)k k! ( 2t et + 1 )w ext = ( ∞∑ n=0 { n+ r k + r } r tn n! )( ∞∑ n=0 Gw n (x) tn n! ) = ∞∑ n=0 n∑ j=0 { j + r k + r } r tj j! Gw n−j(x) tn−j (n− j)! = ∞∑ n=0  n∑ j=0 { j + r k + r } r n! (n− j)!j! Gw n−j(x)  tn n! = ∞∑ n=0  n∑ j=0 { j + r k + r } r ( n j ) Gw n−j(x)  tn n! Comparing the coefficients of tn n! yields the desired convolution formula in (16). Theorem 3.2. The Horizontal Generating Function for both kinds of r-Stirling Genocchi Polynomials of higher order are given as follows: Gw n (x+ z − r) = n∑ k=0 SG1w n (x; k, r)zk (17) 19 of 26 Gw n (x+ z + r) = n∑ k=0 SG2w n (x; k, r)zk (18) Proof. The exponential generating function of (13) can be written as ∞∑ k=0 { ∞∑ n=k SG1w n (x; k, r) tn n! } zk = ( 1 1 + t )r ( 2t et + 1 )w ext ∑ k≥0 lnk(1 + t) k! zk = ( 1 1 + t )r ( 2t et + 1 )w ext ∑ k≥0 [z ln(1 + t)]k k! = ( 1 1 + t )r eln(1+t)z ( 2t et + 1 )w ext = ∑ n≥0 ( z − r n ) tn ∞∑ n=0 Gw n (x) tn n! = ∑ n≥0 (z − r)n tn n! ( ∞∑ n=0 Gw n (x) tn n! ) = ∞∑ n=0 n∑ m=0 (z − r)m tm m! Gw n−m(x) tn−m (n−m)! = ∞∑ n=0 { n∑ m=0 (z − r)m 1 (n−m)!m! Gw n−m(x) } tn = ∞∑ n=0 { n∑ m=0 (z − r)m ( n m ) Gw n−m(x) } tn n! = ∞∑ n=0 { n∑ m=0 ( n m ) Gw n−m(x)(z − r)m } tn n! Rewriting ∞∑ n=0 { n∑ k=0 SG1w n (x; k, r)zk } tn n! = ∞∑ n=0 { n∑ m=0 ( n m ) Gw n−m(x)(z − r)m } tn n! Comparing the coefficients of tn n! , we have n∑ k=0 SG1w n (x; k, r)zk = n∑ m=0 ( n m ) Gw n−m(x)(z − r)m Applying the Addition Formula, we have n∑ k=0 SG1w n (x; k, r)zk = Gw n (x+ z − r) 20 of 26 Similarly (14) can be written as ∞∑ k=0 { ∞∑ n=k SG2w n (x; k, r) tn n! } zk = ∞∑ k=0 {( 2t et + 1 )w ext ert(et − 1)k k! } zk = ( 2t et + 1 )w extert ∞∑ k=0 ( z k ) (et − 1)k = ( 2t et + 1 )w extert(1 + (et − 1))z = e(z+r)t ( 2t et + 1 )w ext = ( ∞∑ n=0 (z + r)n tn n! )( ∞∑ n=0 Gw n (x) tn n! ) = ∞∑ n=0 n∑ m=0 (z + r)m tm m! Gw n−m(x) tn−m (n−m)! = ∞∑ n=0 { n∑ m=0 (z + r)m 1 (n−m)!m! Gw n−m(x) } tn = ∞∑ n=0 { n∑ m=0 (z + r)m ( n m ) Gw n−m(x) } tn n! = ∞∑ n=0 { n∑ m=0 ( n m ) Gw n−m(x)(z + r)m } tn n! Rewriting ∞∑ n=0 { n∑ k=0 SG2w n (x; k, r)zk } tn n! = ∞∑ n=0 { n∑ m=0 ( n m ) Gw n−m(x)(z + r)m } tn n! Comparing the coefficients of tn n! , we have n∑ k=0 SG2w n (x; k, r)zk = n∑ m=0 ( n m ) Gw n−m(x)(z + r)m Applying the Addition Formula, we have n∑ k=0 SG2w n (x; k, r)zk = Gw n (x+ z + r) Hence, we proved the horizontal generating function in (17) and (18). 21 of 26 Theorem 3.3. The formula for the first kind of r-Stirling Genocchi Polynomials of higher order is explicitly stated as follows: SG1w n (x; k, r) = n∑ m=0 n∑ j=m m−k∑ f=0 f∑ d=i (−1)n−j+d+fGw j−m(x) ( j m )( n j )( f d ) ( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! rn−j Proof. The exponential generating function in (13) composed of three functions. The first function can be expressed as( 1 1− t )r = ∑ n≥0 (−1)nrn tn n! where rn = r(r + 1) . . . (r + n− 1) The second function can be expresses as the Genocchi Polynomial of higher order,( 2t et + 1 )w ext = ∞∑ n=0 Gw n (x) tn n! Lastly the third function can be expressed 1 k! [ln (1 + t)]k = ∑ n≥k s(n, k) tn n! Hence, using Cauchy’s Rule for the product of power series, we have n∑ k≥0 SG1w n (x; k, r) tn n! = ∑ n≥0 (−1)nrn tn n! ( ∞∑ n=0 Gw n (x) tn n! )∑ n≥k s(n, k) tn n!  = ∑ n≥0 (−1)nrn tn n! ( ∞∑ n=0 { n∑ m=k s(m, k) ( n m ) Gw n−m(x) } tn n! ) = ∞∑ n=0 n∑ j=0 j∑ m=k s(m, k) ( j m ) Gw j−m(x) tj j! (−1)n−jrn−j tn−j (n− j)! = ∞∑ n=0 n∑ j=0 j∑ m=k s(m, k) ( j m ) Gw j−m(x) tj−j+n (n− j)!j! (−1)n−jrn−j = ∞∑ n=0  n∑ m=0 n∑ j=m s(m, k) ( j m ) Gw j−m(x) 1 (n− j)!j! (−1)n−jrn−j  tn = ∞∑ n=0  n∑ m=0 n∑ j=m s(m, k) ( j m ) Gw j−m(x) ( n j ) (−1)n−jrn−j  tn n! 22 of 26 Comparing the coefficients of tn n! , SG1w n (x; k, r) = n∑ m=0 n∑ j=m s(m, k) ( j m ) Gw j−m(x) ( n j ) (−1)n−jrn−j Using the Schlömilch Formula for the Stirling numbers of the first kind, s(n, k) = n−k∑ r=0 r∑ j=i (−1)j+r ( r j )( n− 1 + r n− k + r )( 2n− k n− k + r ) (r − j)n−k+r r! Then the Schlömilch Formula for the r-Stirling Genocchi number of the first kind is given by SG1w n (x; k, r) = n∑ m=0 n∑ j=m ( j m ) Gw j−m(x) ( n j ) (−1)n−jrn−j m−k∑ f=0 f∑ d=i (−1)d+f ( f d )( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! = n∑ m=0 n∑ j=m m−k∑ f=0 f∑ d=i ( j m ) Gw j−m(x) ( n j ) (−1)n−jrn−j (−1)d+f ( f d )( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! = n∑ m=0 n∑ j=m m−k∑ f=0 f∑ d=i (−1)n−j+d+fGw j−m(x) ( j m )( n j )( f d )( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! rn−j Theorem 3.4. The formula for the second kind of r-Stirling Genocchi Polynomials of higher order is explicitly defined by the following formula: SG2w n (x; k, r) = n∑ m=0 { m+ r k + r } r ( n m ) Gw n−m(x) Proof. The exponential generating function in (14) can be written as n∑ n≥k k!SG2w n (x; k, r) tn n! = ( 2t et + 1 )w extert k∑ i=0 ( k i ) (et)k−i(−1)i 23 of 26 = ( 2t et + 1 )w ext k∑ i=0 ( k i ) ert(et)k−i(−1)i = ( 2t et + 1 )w ext k∑ i=0 ( k i ) ert+(k−i)t(−1)i = k∑ i=0 ( k i ) ert(et)k−i(−1)i ∞∑ n=0 Gw n (x) tn n! =  k∑ i=0 ( k i )∑ n≥0 [((k − i) + r)t]n n! (−1)i ( ∞∑ n=0 Gw n (x) tn n! ) =  ∞∑ n≥0 { k∑ i=0 (−1)i ( k i ) ((k − i) + r)n } tn n! ( ∞∑ n=0 Gw n (x) tn n! ) =  ∞∑ n≥0 k! { n+ r k + r } r tn n! ( ∞∑ n=0 Gw n (x) tn n! ) = ∞∑ n=0 n∑ m=0 k! { m+ r k + r } r tm m! Gw n−m(x) tn−m (n−m)! = ∞∑ n=0 { n∑ m=0 k! { m+ r k + r } r 1 (n−m)!m! Gw n−m(x) } tn = ∞∑ n=0 { n∑ m=0 k! { m+ r k + r } r ( n m ) Gw n−m(x) } tn n! Comparing the coefficients of tn n! completes the proof of the theorem. When x = 0 in the Genocchi polynomials of higher order , we have the following corollaries. Corollary 3.5. Convolution Formula of the r-Stirling Numbers and Genocchi numbers of higher order are given as follows: SG1w n (k; r) = n∑ j=k ̂[j + r k + r ] r ( n j ) Gw n−j SG2w n (k; r) = n∑ j=k { j + r k + r } r ( n j ) Gw n−j where n ≥ k. Proof. Setting x = 0 of Theorem 9, the proof of this theorem follows immediately. 24 of 26 Corollary 3.6. The Horizontal Generating Function for both kinds of r-Stirling Genocchi Numbers of higher order are given as follows: Gw n (z − r) = n∑ k=0 SG1w n (k, r)zk Gw n (z + r) = n∑ k=0 SG2w n (k, r)zk Proof. Setting x = 0 of Theorem 10, the proof of this theorem follows immediately. Corollary 3.7. The formula for the first kind of r-Stirling Genocchi Numbers of higher order is explicitly stated as follows: SG1w n (k; r) = n∑ m=0 n∑ j=m m−k∑ f=0 f∑ d=i (−1)n−j+d+fGw j−m ( j m )( n j )( f d ) ( m− 1 + f m− k + f )( 2m− k m− k + f ) (f − d)m−k+f f ! rn−j Proof. Setting x = 0 of Theorem 11, the proof of this theorem follows immediately. Corollary 3.8. The formula for the second kind of r-Stirling Genocchi Numbers of higher order is explicitly defined by the following formula: SG2w n (k; r) = n∑ m=0 { m+ r k + r } r ( n m ) Gw n−m Proof. Setting x = 0 of Theorem 12, the proof of this theorem follows immediately. 4. Conclusion and Recommendation In this research, we introduced the novel concept of r-Stirling Genocchi numbers, ex- panding the field’s understanding of combinatorial number theory by drawing connections between r-Stirling and Genocchi numbers. We derived significant results that include a convolution formula, which establishes structural relations among these numbers, as well as a horizontal generating function, which provides insight into the sequence’s behavior and recursive properties. Additionally, we formulated explicit expressions for both the first and second kinds of r-Stirling Genocchi numbers, offering concrete tools for calculating these values directly. The study was also extended to encompass Genocchi polynomials and higher-order Genocchi polynomials, broadening the scope of applicability. These findings contribute to the theoretical framework and may open up avenues for further exploration into generalized Stirling and Genocchi number applications, particu- larly in areas involving combinatorial identities, partition theory, and potentially in solving 25 of 26 specific recurrence relations. Furthermore, the results of this study, particularly the use of exponential generating functions, convolution identities, and explicit formulas parallel the methods employed in recent work on degenerate r-Whitney numbers and polynomials in [10] . This alignment suggests that the framework developed here can be naturally extended to define and investigate r-Whitney Genocchi numbers, potentially including degenerate versions. Acknowledgements The authors sincerely thank the referees for their thorough and insightful review of the manuscript. 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