EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6743 ISSN 1307-5543 – ejpam.com Published by New York Business Global Integral Formulas for the Noncentral Tanny-Dowling Polynomials Mahid M. Mangontarum1,2,∗, Norlailah M. Madid1, Asnawi A. Campong1 1 Department of Mathematics, Mindanao State University-Main Campus, Marawi City 9700, Philippines 2 Mamitua Saber Institute of Research and Creation, Mindanao State University Main Campus, Marawi City 9700, Philippines Abstract. In this paper, the authors established some integral formulas for the noncentral Tanny- Dowling polynomials. These formulas are shown to be generalizations of some known results on the classical geometric polynomials. 2020 Mathematics Subject Classifications: 11B83, 11B73 Key Words and Phrases: Geometric polynomial, exponential polynomial, noncentral Tanny- Dowling polynomial, noncentral Dowling polynomial 1. Introduction Let { n k } denote the Stirling numbers of the second kind, see [1]. In the classical distribution problems, { n k } count the number of ways to distribute n distinct objects into k identical boxes such that no box is empty, see page 47 of [2]. These numbers also appear as coefficients in the expansion of xn = n∑ k=0 { n k } (x)k, (1) where (x)k = x(x− 1)(x− 2) · · · (x− k + 1) is the Pochhammer symbol, see [3]. It is easy to see that k! { n k } i.e., the Stirling numbers of the second kind multiplied by k!, counts the number of ways to distribute n distinct objects to k distinct boxes such that no box is empty. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6743 Email addresses: mangontarum.mahid@msumain.edu.ph (M. M. Mangontarum), norlailah.madid@msumain.edu.ph (N. M. Madid), campong.aa82@s.msumain.edu.ph (A. A. Campong) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. M. Mangontarum et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6743 2 of 8 The geometric polynomials, also known as Fubini Polynomials, see [4], are defined by wn(x) = n∑ k=0 k! { n k } xk. (2) These polynomials are known to satisfy the exponential generating function given by [5, Eq. (3.14)] ∞∑ n=0 wn(x) zn n! = 1 1− x(ez − 1) . (3) These polynomials have strong links to combinatorics, exponential generating functions, and classical sequences such as the Bernoulli numbers. The case when x = 1 given by wn := wn(1) = n∑ k=0 k! { n k } (4) is called geometric numbers or Fubini numbers. These count all the possible set partitions of an n element set such that the order of the blocks matters. The exponential generating function of wn can be easily by setting x = 1 in (3). That is, ∞∑ n=0 wn(1) zn n! := ∞∑ n=0 wn zn n! = 1 2− ez . (5) The study of geometric polynomials has remained a thrend for among mathematicians to this date. For instance, Kellner [6] established several identities involving the polyno- mials wn(x). Among these identities is the integral identity over the interval [−1, 0] given by ∫ 0 −1 wn(x)dx = Bn. (6) Here, Bn denotes the nth Bernoulli number defined by the exponential generating function ∞∑ n=0 Bn xk n! = x ex − 1 . (7) The proof of (6) uses Worpitzky’s identity [7, pg. 215 (36)] given by Bn = n∑ k=0 k∑ j=0 (−1)j ( k j ) jn k + 1 (8) and its equivalent form Bn = n∑ k=1 (−1)k k! k + 1 { n k } . (9) M. M. Mangontarum et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6743 3 of 8 Boyadzhiev [5] established transformation formulas for the geometric polynomials. In his paper, given the exponential polynomials or Bell polynomials ϕn(x) defined by ϕn(x) = n∑ k=0 { n k } xk, (10) Boyadzhiev [5] expressed the geometric polynomials wn(x) in terms of the exponential polynomials, as follows wn(x) = ∫ ∞ 0 ϕn(xλ)e −λdλ. (11) This was used to derive more properties for wn(x) including the exponential generating function [5, Eq. (3.13)] ∫ ∞ 0 e−λ(1−x(ex−1))dλ = ∞∑ n=0 wn(x) zn n! . (12) Additional important works are due to Kargın [8], Dil and Kurt [9], Boyadzhiev and Dil [10], Kargın and Çekim [11], Ramı́rez and Cesarano [12], among others. In 2016, Mangontarum et al. [13] introduced the noncentral Tanny-Dowling polyno- mials F̃m,a(n;x) defined by F̃m,a(n;x) = n∑ k=0 k!W̃m,a(n, k)x k (13) and satisfying the exponential generating function ∞∑ n=k F̃m,a(n;x) zn n! = me−az m− x(emz − 1) , (14) where the numbers W̃m,a(n, k) are the noncentral Whitney numbers of the second kind, a generalization of { n k } . The parameters (m, a) deform the classical structure: F̃1,0(n;x) = wn(x). Further, in a recent paper by Mangontarum and Madid [14], a number of identities for F̃m,a(n;x) are established. Such identities are shown to generalize some known results on the geometric polynomials, including the ones in the paper of Kargın [8]. In the present paper, the authors establish integral formulas for and involving the noncentral Tanny-Dowling polynomials. In particulay, we will derive a generalization of Kellner’s [6] integral formula relating the noncentral Tanny-Dowling polynomials with the Bernoulli polynomials, obtain generalizations of the Worpitzky’s [7] explicit formulas in terms of noncentral Whitney numbers of the second kind, and derive a generalizations of Boyadzhiev’s [5] identities for the noncentral Tanny-Dowling polynomials. M. M. Mangontarum et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6743 4 of 8 2. Results and Discussions The first theorem establishes a relationship between the noncentral Tanny-Dowling polynomials and the Bernoulli polynomials, and extends the result of Kellner [6] presented in (6). Theorem 1. For any real number a and positive integer m, the following integral formula over the interval [−1, 0] holds:∫ 0 −1 F̃m,a(n;mx)dx = mnBn ( −a m ) . (15) Proof. Note that from (14), we have ∞∑ n=0 ∫ 0 −1 F̃m,a(n;mx) zn n! dx = ∫ 0 −1 e−az 1− xemz + x dx. (16) Evaluating the integral and by (7), we get ∞∑ n=0 ∫ 0 −1 F̃m,a(n;mx) zn n! dx = −e−az emz − 1 ln |1− xemz + x| ∣∣∣∣0 −1 = −e−az emz − 1 (− ln |emz|) = mze−az emz − 1 = ∞∑ n=0 mnBn ( −a m ) zn n! . Comparing the coefficients of zn n! completes the proof. Remark 1. Since F̃1,0(n;x) = wn(x), then by setting m = 1 and a = 0 in Theorem 1, we get the integral ∫ 0 −1 F̃1,0(n;x)dx = Bn(0) which is Kellner’s [6] identity in (6). Now, observe that from (13), mnBn ( −a m ) = ∫ 0 −1 ( n∑ k=0 mkk!W̃m,a(n, k)x k ) dx = n∑ k=0 mkk!W̃m,a(n, k) ∫ 0 −1 xkdx M. M. Mangontarum et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6743 5 of 8 = n∑ k=0 mkk!W̃m,a(n, k) (−1)k k + 1 . Thus, we have the following corollary: Corollary 1. The nth Bernoulli polynomial Bn (−a m ) satisfies the following explicit for- mula: Bn ( −a m ) = n∑ k=0 k!W̃m,a(n, k) (−1)k mn−k(k + 1) . (17) Remark 2. Since W̃1,0(n, k) = { n k } , then when m = 1 and a = 0 in Corollary 1, we recover the Bernoulli formula in (9). Moreover, using the explicit formula of W̃m,a(n, k) [13, Eq. (38)] given by W̃m,a(n, k) = 1 mkk! k∑ j=0 ( k j ) (−1)k−j(mj − a)n, equation (17) can be written as Bn ( −a m ) = n∑ k=0 k∑ j=0 ( k j ) (mj − a)n (−1)j mn(k + 1) . (18) This is a generalization of Worpitzky’s [7] identity in (8). Before proceeding, note that by induction on k, it is easy to show that∫ ∞ 0 xke−xdx = k!. (19) Also, the noncentral Dowling polynomials [13, Eq. (89)] defined by D̃m,a(n;x) = n∑ k=0 W̃m,a(n, k)x k satisfies the exponential generating function [13, Eq. (91)] ∞∑ n=0 D̃m,a(n;x) zn n! = e−az+(emz−a)(x/m). (20) The next theorem provides an integral representation of the noncentral Tanny-Dowling polynomials in terms of noncentral Dowling polynomials. This extends Boyadzhiev’s [5] identity for geometric polynomials in (11). M. M. Mangontarum et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6743 6 of 8 Theorem 2. The noncentral Tanny-Dowling polynomials satisfy the following relation: F̃m,a(n;x) = ∫ ∞ 0 D̃m,a(n;xλ)e −λ dλ. (21) Proof. By definition,∫ ∞ 0 D̃m,a(n;xλ)e −λdλ = ∫ ∞ 0 [ n∑ k=0 W̃m,a(n, k)x kλk ] e−λdλ = n∑ k=0 W̃m,a(n, k)x k ∫ ∞ 0 λke−λdλ. Using (19) and then (13) yield∫ ∞ 0 D̃m,a(n;xλ)e −λdλ = n∑ k=0 k!W̃m,a(n, k)x k = F̃m,a(n;x) which is the desired result. Remark 3. Since D̃1,0(n;xλ) = ϕn(xλ), then when m = 1 and a = 0, the following relation ∫ ∞ 0 D̃1,0(n;xλ)e −λ dλ = F̃1,0(n;x) (22) is precisely Boyadzhiev’s [5] formula in (11). Finally, the next theorem presents another form of exponential generating function for the polynomials F̃m,a(n;x). Theorem 3. The exponential generating function of the noncentral Tanny-Dowling poly- nomial satisfies the following integral formula: ∞∑ n=0 F̃m,a(n;x) zn n! = ∫ ∞ 0 exp [ −az − λ ( 1− x m (emz − 1) )] dλ. (23) Proof. Multiplying both sides of (21) by zn n! and summing over n gives ∞∑ n=0 F̃m,a(n;x) zn n! = ∫ ∞ 0 ( e−λ ∞∑ n=0 D̃m,a(n;xλ) zn n! ) dλ. By apply (20) in the right-hand side, ∞∑ n=0 F̃m,a(n;x) zn n! = ∫ ∞ 0 e−az−λ(1− x m (emz−1))dλ. M. M. Mangontarum et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6743 7 of 8 Remark 4. When m = 1 and a = 0, we obtain∫ ∞ 0 e−(0)z−λ(1−x 1 (e(1)z−1)dλ = ∞∑ n=0 F̃1,0(n;x) zn n! , (24) an equivalent representation of Boyadzhiev’s [5] exponential generating function in (12). 3. Conclusion The results of this study demonstrate the relationship between the noncentral Tanny- Dowling polynomials, a natural generalization of the Bell polynomials, and the Bernoulli polynomials. It is interesting to explore similar connections between the noncentral Tanny- Dowling polynomials and other families of special polynomials discussed in [12], such as the Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials. 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