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axioms Article Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function 1 2 3, Irem Kucukoglu , Burcin Simsek and Yilmaz Simsek * Department of Engineering Fundamental Sciences, Faculty of Engineering, Alanya Alaaddin Keykubat University, TR-07425 Antalya, Turkey; irem.kucukoglu@alanya.edu.tr Department of Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USA; bus5@pitt.edu Department of Mathematics, Faculty of Science University of Akdeniz, TR-07058 Antalya, Turkey * Correspondence: ysimsek@akdeniz.edu.tr Received: 14 September 2019; Accepted: 9 October 2019; Published: 11 October 2019 Abstract: The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefﬁcients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefﬁcients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution. Keywords: generating functions; functional equations; partial differential equations; special numbers and polynomials; Bernoulli numbers; Euler numbers; Stirling numbers; Bell polynomials; Cauchy numbers; Poisson-Charlier polynomials; Bernstein basis functions; Daehee numbers and polynomials; combinatorial sums; binomial coefﬁcients; p-adic integral; probability distribution MSC: Primary 05A10; 05A15; 11B73; 11B68; 11B83; Secondary 05A19; 11B37; 11S23; 26C05; 34A99; 35A99; 40C10 1. Introduction In recent years, generating functions and their applications on functional equations and differential equations has gained high attention in various areas. These techniques allow researchers to derive various identities and combinatorial sums that yield important special numbers and polynomials. In fact, the current trend is to combine the p-adic integrals with these techniques. In most of ﬁelds of mathematics and physics, different applications of generating functions are used as an important tool. For instance, a common research topic in quantum physics is to identify a generating function that could be a solution to a differential equation. The motivation of this paper is to outline the advantages of techniques associated with generating functions. First, generating functions are presented for new families of combinatorial numbers and polynomials. Second, we derive new identities, relations, and formulas including the Bersntein basis functions, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Axioms 2019, 8, 112; doi:10.3390/axioms8040112 www.mdpi.com/journal/axioms Axioms 2019, 8, 112 2 of 16 Poisson–Charlier polynomials, the Daehee numbers and polynomials, the probability distribution functions, as well as combinatorial sums including the Bernoulli numbers, the Euler numbers, the Cauchy numbers (or the Bernoulli numbers of the second kind), and combinatorial numbers. With the followings, we brieﬂy introduce the notations, deﬁnitions, relations, and formulas are used throughout this paper: As usual, let N, Z, N , Q, R, and C denote the set of natural numbers, set of integers, set of nonnegative integers, set of rational numbers, set of real numbers, and set of complex numbers, respectively. Let log z denote the principal branch of the multi-valued function log z with the imaginary part Im(log z) constrained by the interval (p, p]. We also assume that: 1, if n = 0 0 = 0, if n 2 N. Moreover, z z(z 1) (z v + 1) (z) = = (v 2 N, z 2 C) v v! v! so that, = z = 1 ( ) (cf. [1–31]). The Poisson–Charlier polynomials C x; a , which are members of the family of Sheffer-type ( ) sequences, are deﬁned as below: ¥ n t t F (t, x; a) = e + 1 = C (x; a) , (1) pc å n a n! n=0 where, (x) nj C (x; a) = (1) , (2) j a j=0 (cf. [16], (p. 120, [18]), [24]). Let x 2 [0, 1] and let n and k be nonnegative integers. The Bernstein basis functions, B (x), are deﬁned by: n k nk B (x) = x (1 x) , (k = 0, 1, . . . , n) (3) so that, n n! k k!(n k)! and its generating function is given by: k ¥ (1x)t n (xt) e t F (t, x; k) = = B (x) , (4) B å k! n! n=0 where t 2 C (cf. [1,15,20,26]). The Stirling numbers of the ﬁrst kind, S (n, k), are deﬁned by the following generating function: ¥ n (log (1 + t)) t F (t; k) = = S (n, k) , (k 2 N ) (5) S å 1 0 k! n! n=k so that, (x) = S (n, k) x (6) å 1 k=0 Axioms 2019, 8, 112 3 of 16 (cf. [2–4,29,30]; see also the references cited therein). The l-Stirling numbers of the second kind, S (n, k; l), are deﬁned with generating function given below (cf. [21,30]): t ¥ le 1 t F (t; v; l) = = S (n, v; l) , (v 2 N ). (7) S å 2 0 v! n! n=0 Notice here that, when l = 1, this reduces to the Stirling numbers of the second kind, S (n, v), whose generating function is given below: t ¥ e 1 t F (t; v) = = S (n, v) , (v 2 N ), (8) S å 2 0 v! n! n=0 namely, S (n, v) = S (n, v; 1) (cf. [2,5,21,30]). 2 2 The Bell polynomials (i.e., exponential polynomials), Bl (x), is deﬁned by: Bl (x) = S (n, v) x (9) å 2 v=1 so that the generating function for the Bell polynomials is given by: ¥ n e 1 x ( ) F (t, x) = e = Bl (x) (10) Bell å n! n=0 (cf. [4,18]). (k) (k) The numbers Y (l) and the polynomials Y (x; l) are deﬁned by the following generating n n functions, respectively: ¥ n 2 t (k) F (t, k; l) = = Y (l) , (11) å n l (1 + lt) 1 n! n=0 and, ¥ n (k) F (t, x, k; l) = F (t, k; l) (1 + lt) = Y (x; l) , (12) n! n=0 where k 2 N and l is real or complex number (cf. [14]). Substituting k = 1 into Equation (11), we have: (1) Y (l) = Y (l) n n (cf. [23]). Substituting k = 1 and l = 1 into Equation (11), we get the following well-known relation between the numbers Y (l) and the Changhee numbers of the ﬁrst kind, Ch : n n n+1 Ch = (1) Y (1). n n Thus we have, (1) n! Ch = = S (n, k)E (13) n 1 å k n + 1 k=0 where the Changhee numbers of the ﬁrst kind, Ch are deﬁned means of the following generating function: Axioms 2019, 8, 112 4 of 16 ¥ n 2 t = Ch (14) t + 1 n! n=0 (cf. [9], see also [7]). The Daehee polynomials, D (x), is deﬁned by the following generating functions (cf. [8]): ¥ n log (1 + t) t F x, t = 1 + t = D x (15) ( ) ( ) ( ) D å n t n! n=0 which, for x = 0, corresponds the generating functions of the Daehee number, D = D (0), given by n n the following explicit formula: (1) n! D = . (16) n + 1 The combinatorial numbers, y (n, k; l), are deﬁned by the following generating function: 1 t F (t, k; l) = le + 1 = y (n, k; l) (17) y å 1 k! n! n=0 where k 2 N and l 2 C (cf. [22]). Use the preceding generating function for the combinatorial numbers, y (n, k; l) to compute the following explicit formula: 1 k j n y n, k; l = l j (18) ( ) k! j j=0 (cf. [22] (Theorem 1, Equation (9))). Note that the following equality holds true: 1 d k y (n, k; l) = le + 1 (19) k! dt t=0 (cf. [31] (p. 64)). When l = 1, if we multiply the numbers y (n, k; l) by k!, then Equation (18) is reduced to the following combinatorial numbers (cf. [6,19,22]): B(k, n) = j j=0 which satisﬁes the following differential equation: B (k, n) = e + 1 (20) dt t=0 (cf. [6], (Equation (2), p. 2 [22])). The combinatorial numbers B(n, k) have various kinds of combinatorial applications. For instance, Ross [19] (pp. 18–20, Exercises 10–12) gave the following applications for solutions of exercises 10–12: From a group of n people, suppose that we want to choose a committee of k, k n, one of whom is to be designated as chairperson. How many different selections are there in which the chairperson and the secretary are the same? n1 Ross [19] (p. 18, Exercise 12) gave the following answer: B(1, n) = n2 . By using the preceding idea summarized above, the following combinatorial identities are obtained: n1 k = n2 k=0 Axioms 2019, 8, 112 5 of 16 2 n2 k = 2 n(n + 1) k=0 and, 3 n3 2 k = 2 n (n + 3) k=0 (cf. (pp. 18–20, Exercises 10–12 [19]), [22,25]). Observe that these numbers are also arised from Equation (20). Next, we present the outline of the present paper: In Section 2, we construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions, we not only investigate properties of these new families, but also provide some new identities and relations with the inclusion of the Bersntein basis functions, combinatorial numbers, and the Stirling numbers. In Section 3, we obtain some derivative formulas and recurrence relations for these new families of combinatorial numbers and polynomials by using differential equations that are a result of these generating functions and their partial derivatives. In Section 4, by using functional equations of the generating functions, we derive some formulas and combinatorial sums including binomials coefﬁcients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, and the Bersntein basis functions. In Section 5, by applying the p-adic integrals and Riemann integral to some new formulas derived by the authors of this paper, some combinatorial sums comprising the binomial coefﬁcients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e. exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind) are presented. In Section 6, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution. In Section 7, we conclude our ﬁndings. 2. New Families of the Combinatorial Numbers and Polynomials In this section, we deﬁne new families of the combinatorial numbers and polynomials by the following generating functions, respectively: ¥ n (k) G (t, k; l) = 2 (l (1 + lt) 1) = Y (l) (21) å n n! n=0 and, G (t, x, k; l) = G (t, k; l) (1 + lt) = Q (x; l, k) (22) n! n=0 where k 2 N and l is a real or complex number. Combining Equations (21) and (22), we get: ¥ n ¥ n n t n t (k) nj Q (x; l, k) = l Y (l) (x) . (23) å å å j nj n! j n! n=0 n=0 j=0 Comparing coefﬁcient of on both sides of the above equation, we arrive at the n! following theorem: Theorem 1. ( ) nj Q (x; l, k) = l Y (l) (x) . (24) nj j=0 Axioms 2019, 8, 112 6 of 16 By the binomial theorem, we have: ¥ n ¥ t k (k) kn k 2n n Y (l) = 2 l (l 1) t . å n å n! n n=0 n=0 Comparing the coefﬁcient of t on both sides of the above equation, we arrive at the following theorem: Theorem 2. Let k and n be nonnegative integers. Then: k kn k 2n 2 n! l l 1 if n k (k) ( ) ( ) Y (l) = (25) 0 if n > k. (k) By Equation (25), a few values of the numbers Y (l) are computed as follows: (k) Y (l) = 2 (l 1) , (k) k 2 k1 Y (l) = 2 l (l 1) , (k) k 4 k2 Y l = 2 2! l l 1 , ( ) ( ) (k) k 2j kj Y l = 2 j! l l 1 for j k, ( ) ( ) (k) k 2k Y (l) = 2 k!l , (k) Y (l) = 0 for j > k. By Equations (24) and (25), we also compute a few values of the polynomials Q x; l, k as follows: ( ) Q (x; l, k) = 2 (l 1) , k k1 k k 2 Q (x; l, k) = 2 (l 1) lx + 2 kl (l 1) , k k k1 k 2 2 k 2 k+1 3 Q (x; l, k) = 2 (l 1) l x + 2 (l 1) l + 2 kl (l 1) x k 4 k1 +2 k (k 1) l (l 1) . (k) By Equation (3), we arrive at a computation formula, for the numbers Y (l), in terms of the Bernstein basis functions by the following corollary: Corollary 1. Let n and k be nonnegative integers and l 2 [0, 1]. Then, kn k n k 2 n! (1) l B (l) if n k (k) Y (l) = (26) 0 if n > k. log(1+lt) Replacing 1 + lt by e leads Equation (21) to be: ¥ k t 1 ( ) (k) log(1+lt) Y (l) = le + 1 . (27) å n n! n=0 Axioms 2019, 8, 112 7 of 16 By combining Equation (17) with the above quation, we get: k m ¥ n ¥ t (1) k! (log (1 + lt)) (k) Y (l) = y (m, k;l) (28) n 1 å å n! m! n=0 m=0 which follows from Equation (5) that: ¥ n ¥ n n t (1) k! t (k) Y (l) = l y (m, k;l) S (n, m) . (29) å n å å 1 1 n! 2 n! n=0 n=0 m=0 Therefore, by comparing coefﬁcient of on both sides of the above equation, we arrive at the n! following theorem: Theorem 3. (1) k! (k) Y l = l y m, k;l S n, m . (30) ( ) ( ) ( ) n 1 1 m=0 Combining Equations (25) with (30) yields the following corollary: Corollary 2. k kn n (1) l (l1) if n k (kn)! y (m, k;l) S (n, m) = (31) å 1 1 0 if n > k. m=0 If we also combine Equations (26) with (30), then we have the following result: Corollary 3. Let n and k nonnegative integer with n k. Then, n! y (m, k;l) S (n, m) = (1) B (l). (32) å 1 1 n k! m=0 On the other hand, since the following equality holds true (cf. [13]): S (n, k; l) = (1) y (n, k;l) , (33) Equation (31) leads the following corollary: Corollary 4. kn n l (l1) if n k (kn)! S (m, k; l) S (n, m) = (34) å 2 1 0 if n > k. m=0 3. Derivative Formulas and Recurrence Relations Arising from Differential Equations of Generating Functions In this section, by using differential equations involving the generating functions G (t, k; l) and G (t, x, k; l) and their partial derivatives with respect to the parameters t, l, and x, we (k) obtain some derivative formulas and recurrence relations for the numbers Y (l) and the polynomials Q (x; l, k). Differentiating both sides of Equation (21) with respect to l, we get the following partial derivative equation: ¶ k fG (t, k; l)g = (2lt + 1)G (t, k 1; l) . (35) ¶l 2 Axioms 2019, 8, 112 8 of 16 Also, if we differentiate both sides of Equation (21) with respect to t, then we get the following partial derivative equation: ¶ kl fG (t, k; l)g = G (t, k 1; l) . (36) ¶t 2 By combining Equation (35) with the RHS of Equation (21), we obtain: ¥ n ¥ n d t k t (k) (k+1) (k+1) fY (l)g = 2nlY (l) + Y (l) . (37) n n å å n1 dl n! 2 n! n=0 n=0 Comparing the coefﬁcients of on both sides of the above equation, we arrive at the n! following theorem: Theorem 4. Let n 2 N. Then, we have: d k (k) (k+1) (k+1) fY (l)g = 2nlY (l) + Y (l) . (38) n n n1 dl 2 By combining Equation (36) with the RHS of Equation (21), we get: ¥ n 2 ¥ n ¶ t kl t (k) (k+1) Y (l) = Y (l) . (39) n n å å ¶t n! 2 n! n=0 n=0 which, by comparing the coefﬁcients of on both sides of the above equation, yields the n! following theorem: Theorem 5. Let n 2 N . Then, we have: kl (k) (k+1) Y (l) = Y (l) . (40) n+1 Differentiating both sides of Equation (22) with respect to l, we get the following partial derivative equation: ¶ k fG (t, x, k; l)g = (2lt + 1)G (t, x, k 1; l) + xtG (t, x 1, k; l) . (41) ¶l 2 Furthermore, if we differentiate both sides of the Equation (22) with respect to t, then we also get the following partial derivative equation: ¶ kl fG (t, x, k; l)g = G (t, x, k 1; l) + xlG (t, x 1, k; l) . (42) ¶t 2 Additionally, when we differentiate both sides of Equation (22) with respect to x, we also get the following partial derivative equation: fG (t, x, k; l)g = log (1 + lt)G (t, x, k; l) . (43) ¶x By combining Equation (41) with the RHS of Equation (22), we get: ¥ n ¥ n ¥ n ¶ t k t t fQ (x; l, k)g = (2lt + 1) Q (x; l, k 1) + xt Q (x 1; l, k) n n n å å å ¶l n! 2 n! n! n=0 n=0 n=0 Axioms 2019, 8, 112 9 of 16 which yields: ¥ n ¥ n ¶ t k t fQ (x; l, k)g = (2nlQ (x; l, k 1) + Q (x; l, k 1)) n n å å n1 ¶l n! 2 n! n=0 n=0 +x nQ (x 1; l, k) . å n1 n! n=0 Comparing the coefﬁcients of on both sides of the above equation, we arrive at the n! following theorem: Theorem 6. Let n 2 N. Then, we have: ¶ k fQ (x; l, k)g = knlQ (x; l, k 1) + Q (x; l, k 1) + xnQ (x 1; l, k) . (44) n n1 n n1 ¶l 2 By combining Equation (42) with the RHS of Equation (22), we get: ¥ n 2 ¥ n ¥ n ¶ t kl t t Q (x; l, k) = Q (x; l, k 1) + xl Q (x 1; l, k) n n n å å å ¶t n! 2 n! n! n=0 n=0 n=0 which, by comparing the coefﬁcients of on both sides of the above equation, yields the n! following theorem: Theorem 7. Let n 2 N . Then, we have: kl Q (x; l, k) = Q (x; l, k 1) + xlQ (x 1; l, k) . (45) n+1 n n By combining Equation (43) with the RHS of Equation (22) and the Taylor series of the function log (1 + lt), we get: ¥ ¥ ¥ n n n n ¶ t l t t n1 fQ (x; l, k)g = (1) Q (x; l, k) . n n å å å ¶x n! n n! n=0 n=1 n=0 Applying the Cauchy product rule to the above equation yields: ¥ n ¥ n j+1 n ¶ t n j!l t fQ (x; l, k)g = t (1) Q (x; l, k) . å å å nj ¶x n! j j + 1 n! n=0 n=0 j=0 Comparing the coefﬁcients of on both sides of the above equation, we arrive at the n! following theorem: Theorem 8. Let n 2 N. Then, we have: n1 j+1 ¶ n 1 j!l fQ (x; l, k)g = n (1) Q (x; l, k) . (46) n å n1j ¶x j j + 1 j=0 Remark 1. Substituting Equation (16) into Equation (46) yields the following formula including Daehee numbers: n1 ¶ n 1 j+1 fQ (x; l, k)g = n l D Q (x; l, k) . (47) å j n1j ¶x j j=0 Axioms 2019, 8, 112 10 of 16 By Equation (15), another form of the partial differential Equation (43) is given by: fG (t, x, k; l)g = ltG (t, k; l) F (x, lt) . (48) ¶x By combining Equation (48) with the RHS of the Equations (15) and (22), we get: ¥ n ¥ n ¥ n ¶ t t t (k) fQ (x; l, k)g = lt Y (l) l D (x) . å n å n å n ¶x n! n! n! n=0 n=0 n=0 Applying the Cauchy product rule to the above equation yields: ¥ ¥ n n n ¶ t n t (k) fQ (x; l, k)g = lt l Y (l) D (x) . å å å j nj ¶x n! j n! n=0 n=0 j=0 Comparing the coefﬁcients of on both sides of the above equation, we arrive at the n! following theorem: Theorem 9. n1 ¶ n 1 (k) j+1 fQ (x; l, k)g = n l Y (l) D (x) . å j nj ¶x j j=0 4. Some Identities and Relations Derived from Functional Equations of Generating Functions In this section, by using functional equations of the aforementioned generating functions, we derive some formulas and combinatorial sums including binomials coefﬁcients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, and the Bersntein basis functions. Now, we set the following functional equation: 1 1 F (t, x; a) = F t, x, k; F t,k; e . (49) pc a a Combining the above equation with the Equations (1), (11), and (12), we get: ¥ ¥ ¥ ¥ n n n n t 1 t 1 t t (k) (k) n C (x; a) = Y x; Y (1) . (50) å å n å n å n! a n! a n! n! n=0 n=0 n=0 n=0 Applying the Cauchy product rule to the above equation yields: ¥ n ¥ n l n t n l 1 1 t (k) (k) nl C (x; a) = (1) Y x; Y . (51) å å å å j lj n! l j a a n! n=0 n=0 l=0 j=0 Therefore, by comparing the coefﬁcient of on both sides of the above equation, we arrive at the n! following theorem: Theorem 10. n l n l 1 1 (k) (k) nl C (x; a) = (1) Y x; Y . (52) å å j lj l j a a l=0 j=0 Moreover, we also set the following functional equation: k k x (1) (xt) t xt F (t,x; k)F (t, x; a) = e + 1 . pc k! a Axioms 2019, 8, 112 11 of 16 Combining the above equation with the Equations (1) and (4) yields: k k n ¥ n ¥ n ¥ ¥ n t t (1) (xt) (xt) x t B (x) C (x; a) = . å å å å k n n! n! k! n! n a n=0 n=0 n=0 n=0 Applying the Cauchy product rule to the above equation yields: ! ! nkj ¥ n ¥ nk n k n x (x) n t (1) x n k t j j B (x)C (x; a) = (n) . å å nj å å k j j n! k! j a n! n=0 j=0 n=0 j=0 Therefore, by comparing the coefﬁcient of on both sides of the above equation, we arrive at the n! following theorem: Theorem 11. nj n nk x (x) n (1) (n) n k j j B (x)C (x; a) = . (53) å nj å k j j k! j a j=0 j=0 Additionaly, we also have the following functional equation: k x (xt) t xt F (t, x; a) e = F (t, x; k) + 1 . pc B k! a Combining the above equation with Equations (1) and (4) yields: k ¥ ¥ ¥ ¥ n n n n (xt) t t t x t n n n (1) C (x; a) (x) = B (x) (1) . å n å å å k! n! n! n! n a n=0 n=0 n=0 n=0 Applying the Cauchy product rule to the above equation yields: 0 1 nj nk ¥ nk n ¥ n n (x) B (x) (1) (n) n k t n t j k k nj @ A x C (x; a) = (1) . å å j å å k! j n! j a n! n=0 n=0 j=0 j=0 Therefore, by comparing coefﬁcient of , we arrive at the following theorem: n! Theorem 12. nj nk nk n (x) B (x) (1) (n) n k n j k k nj x C (x; a) = (1) . (54) å j å k! j j a j=0 j=0 By substituting t ! a e 1 into Equation (1), we also get the following functional equation: t tx F a e 1 , x; a = e F (t,a) . (55) pc Bell By Equations (1) and (10), we thus get: ¥ ¥ n ¥ n e 1 t t n n a C (x; a) = x Bl (a) . (56) å n å å n n! n! n! n=0 n=0 n=0 Axioms 2019, 8, 112 12 of 16 Applying the Cauchy product rule to the above equation and combining Equation (8) with the ﬁnal equation yields: ¥ m m ¥ m m t m t n mj a C x; a S (m, n) = x Bl a . (57) ( ) ( ) n 2 j å å å å m! j m! m=0 n=0 m=0 j=0 Therefore, by comparing the coefﬁcient of , we arrive at the following theorem: m! Theorem 13. m m n mj a C (x; a) S (m, n) = x Bl (a) . (58) å n 2 å j n=0 j=0 5. Some Identities and Relations Arising from the p-adic Integrals and Riemann Integral In this section, by applying the p-adic integrals and Riemann integral to some of our results, we derive some combinatorial sums including the binomial coefﬁcients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Let Z denote a set of p-adic integers. Let f x be a uniformly differentiable function on Z . ( ) p p The Volkenborn integral (or p-adic bosonic integral) of the function f (x) is given by: p 1 f (x)dm (x) = lim f (x), (59) 1 å N!¥ p x=0 where, m (x) = m (x + p Z ) = 1 1 p (cf. [17]; see also [11,12]). It is known that the bosonic p-adic integral of the function f (x) = x gives the Bernoulli numbers as follows (cf. [11,17]): B = x dm (x) (60) n 1 where B denotes the Bernoulli numbers of the ﬁrst kind deﬁned by means of the following generating function: ¥ n t t = B , (t < j2pj) (61) å n e 1 n! n=0 which arise in not only analytic number theory, but also other related areas (cf. [5–31]). The fermionic p-adic integral of the function f (x) is given by (cf. [12]): Z p 1 f x dm x = lim 1 f x (62) ( ) ( ) ( ) ( ) N!¥ x=0 where p 6= 2 and, (1) m (x) = m x + p Z = 1 1 p (cf. [10,12]). Axioms 2019, 8, 112 13 of 16 The fermionic p-adic integral of the function f x = x gives the Euler numbers as follows ( ) (cf. [11]): E = x dm (x) , (63) n 1 where E denotes the Euler numbers of the ﬁrst kind deﬁned by means of the following generating function: 2 t = E , (t < jpj) (64) å n e + 1 n! n=0 (cf. [5–30]). It is known that the following p-adic bosonic and fermionic integral representations for the Poisson–Charlier polynomials hold true (see [24] (Equations (33) and (35), pp. 944–945)): n k! C (x; a) dm (x) = (1) (65) n 1 å (k + 1) a k=0 and, n k! C (x; a) dm (x) = (1) . (66) n 1 å (2a) k=0 By applying the bosonic p-adic integral to Equation (58) and combining the ﬁnal equation with Equations (60) and (65), we arrive at the following theorem: Theorem 14. m n nk m (n) a S (m, n) m (1) = B Bl (a) . (67) å å å mj j k + 1 j n=0 k=0 j=0 By applying the fermionic p-adic integral to Equation (58) and combining the ﬁnal equation with Equations (63) and (66), we also arrive at the following theorem: Theorem 15. m n nk m (n) a S (m, n) m n 2 (1) = E Bl (a) . (68) å å å mj j 2 j n=0 k=0 j=0 Moreover, by integrating Equation (58) with respect to x from 0 to 1, we have: 1 1 Z Z m m n mj a S (m, n) C (x; a) dx = Bl (a) x dx. (69) å 2 å j n=0 j=0 0 0 On the other hand, by integrating Equation (2) with respect to x from 0 to 1, we also have: 1 1 Z Z n 1 nj C (x; a) dx = (1) (x) dx, (70) j=0 0 0 By making use of the following deﬁnition of the well-known Cauchy numbers (or the Bernoulli numbers of the second kind) b (0) (cf. [4]): b (0) = (x) dx, (71) 0 Axioms 2019, 8, 112 14 of 16 Equation (70) yields: b (0) nj C (x; a) dx = (1) . (72) j a j=0 Combining the above equation with Equation (69), we arrive at the following theorem: Theorem 16. m n m Bl (a) n m nj nj (1) a S (m, n)b (0) = . (73) å å j å j j m j + 1 n=0 j=0 j=0 6. Applications in the Probability Distribution Function (k) In this section, we investigate some applications of the numbers Y (l). Assume that 0 < p 1 and n = 0, 1, 2, . . . , k. We set the following discrete probability distribution: kn (1) 2 (k) f ( p; k, n) = Y ( p) (74) n! p where p is a probability of success, k is number of trials, n is number of successes in k trials, and n = 0, 1, 2, . . . , k. Therefore, f ( p; k, n) is binomially distributed with parameters (k, p). Properties of Discrete Probability Distribution f p; k, n ( ) Here, we give some properties of discrete probability distribution f ( p; k, n). We examine the properties of the probability distribution f p; k, n with a random variable with parameters k, n, and p ( ) as follows: For all k, n, p with 0 n k and 0 < p 1, 0 f p; k, n 1. That is f p; k, n 0. ( ) ( ) The probability distribution function f ( p; k, n) satisﬁes that: f ( p; k, n) = 1. n=0 Computing the distribution function f ( p; k, n). Suppose that X is a binomial with parameters (k, p). To computing its distribution function: P(X j) = f ( p; k, n) , n=0 where j = 0, 1,. . . ,k. In order to compute its expected value and variance for random variable with parameters k and p: v v E [X ] = n f ( p; k, n) (75) n=0 Observe that the probability distribution function f ( p; k, n) is a modiﬁcation of the binomial probability distribution function with parameters (k, p). Substituting v = 1 into Equation (75), E [X] = k p. Substituting v = 2 into Equation (75), variance E X (E [X]) = k p (1 p). If we take k ! ¥, then the distribution f ( p; k, n) goes to the Poisson distribution. On the other hand the Poisson–Charlier polynomials are orthogonal with respect to the Poisson distribution (cf. [18,24]). Axioms 2019, 8, 112 15 of 16 7. Conclusions Applications of generating functions are used in many areas, and we used them to study new families of combinatorial numbers and polynomials. We then studied properties of these new families, which yielded a handful of new identities and relations. Namely, these identities were related to numerous special numbers, special polynomials, and special functions such as the Bersntein basis functions, the Stirling numbers, the Bell polynomials (or exponential polynomials), the Poisson–Charlier polynomials, and the probability distribution functions. Furthermore, we should note that newly deﬁned combinatorial numbers in this paper gave a different approach to the binomial (or Newton) distribution and the Poisson distribution, as well as combinatorial sums including the Bernoulli numbers, the Euler numbers, the Cauchy numbers (or the Bernoulli numbers of the second kind), and combinatorial numbers. This is why the results of this paper have the potential to be used in numerous areas such as mathematics, probability, physics, and in other associated areas. Author Contributions: Investigation, I.K., B.S., Y.S.; wirting-original draft, I.K., B.S., Y.S.; writing-review and editing, I.K., B.S., Y.S. Funding: This research received no external funding. Acknowledgments: This paper is dedicated to Hari Mohan Srivastava on the occasion of his 80th Birthday. Yilmaz Simsek was supported by the Scientiﬁc Research Project Administration of Akdeniz University. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Acikgoz, M.; Araci, S. On generating function of the Bernstein polynomials. Proc. Int. Conf. Numer. Anal. Appl. Math. Am. Inst. Phys. Conf. Proc. 2010, CP1281, 1141–1143. 2. Bona, M. Introduction to Enumerative Combinatorics; The McGraw-Hill Companies Inc.: New York, NY, USA, 2007. 3. Charalambides, C.A. Enumerative Combinatorics; Chapman and Hall (CRC Press Company): London, UK; New York, NY, USA, 2002. 4. Comtet, L. Advanced Combinatorics: The Art of Finite and Inﬁnite Expansions; D. Reidel Publishing Company: Dordrecht, The Netherlands; Boston, MA, USA, 1974. 5. Djordjevic, G.B.; Milovanovic, ´ G.V. 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Math. 2019, 13, 61–72. [CrossRef] c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Axioms – Multidisciplinary Digital Publishing Institute
Published: Oct 11, 2019
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