Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 March 2017 doi:10.20944/preprints201703.0055.v1 Peer-reviewed version available at Tbilisi Mathematical Journal 2017, 10, 153-158; doi:10.1515/tmj-2017-0053 A CLOSED FORM FOR THE STIRLING POLYNOMIALS IN TERMS OF THE STIRLING NUMBERS FENG QI AND BAI-NI GUO Abstract. In the paper, by virtue of the Fa´adi Bruno formula and two identities for the Bell polynomial of the second kind, the authors find a closed form for the Stirling polynomials in terms of the Stirling numbers of the first and second kinds. 1. Notation and main result It is common knowledge [1, p. 48] that the Bernoulli numbers Bj are defined by 1 1 z X zj z X z2j = B = 1 − + B ; jzj < 2π: ez − 1 j j! 2 2j (2j)! j=0 j=0 For some new developments in recent years about this topic, please refer to [2, 3, 6, 7, 10] and closely-related references therein. The Stirling numbers of the first and second kinds s(n; k) and S(n; k) are im- portant in combinatorial analysis, theory of special functions, and number theory. They can be generated by the rising factorial n−1 n Y X k (x)n = (x + k) = s(n; k)x (1) k=0 k=0 and the exponential function 1 (ex − 1)k X xn = S(n; k) ; k! n! n=k see [1, p. 213, Theorem A] and [1, p. 206, Theorem A], and can be computed by explicit formulas n−1 ` −1 `k−3−1 `k−2−1 X 1 X1 1 X 1 X 1 s(n; k) = (−1)n+k(n − 1)! ··· `1 `2 `k−2 `k−1 `1=1 `2=1 `k−2=1 `k−1=1 for 1 ≤ k ≤ n and k 1 X k S(n; k) = (−1)k−` `n; k! ` `=1 see [9, Corollary 2.3] and [1, p. 206]. For recent investigations on the Stirling numbers s(n; k) and S(n; k), please refer to [6, 7, 8, 9] and plenty of references cited therein. The Stirling polynomials Sk(x) are a family of polynomials that generalize the Bernoulli numbers Bk and the Stirling numbers of the second kind S(n; k). The 2010 Mathematics Subject Classification. Primary 11B83; Secondary 11B68, 33B10. Key words and phrases. closed form; Stirling polynomial; Stirling number; Bernoulli number; Fa´a di Bruno's formula; Bell polynomial. This paper was typeset using AMS-LATEX. 1 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 March 2017 doi:10.20944/preprints201703.0055.v1 Peer-reviewed version available at Tbilisi Mathematical Journal 2017, 10, 153-158; doi:10.1515/tmj-2017-0053 2 F. QI AND B.-N. GUO Stirling polynomials Sk(x) for nonnegative integers k are defined by the generating function x+1 1 t X tk = S (x) : 1 − e−t k k! k=0 The first six Stirling polynomials Sk(x) for 0 ≤ k ≤ 5 are x + 1 3x2 + 5x + 2 x3 + 2x2 + x 1; ; ; ; 2 12 8 15x4 + 30x3 + 5x2 − 18x − 8 3x5 + 5x4 − 5x3 − 13x2 − 6x ; : 240 96 For more information on Sk(x), see the papers [11, 12] and closely-related references therein. In mathematics, a closed form is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, four arithmetic operations, and elementary functions, but usually no limit. In this paper, we find a closed form for the Stirling polynomials Sk(x) in terms of the Stirling numbers s(n; k) and S(n; k) of the first and second kinds. Our main result can be stated as the following theorem. Theorem 1. For k ≥ 0, the Stirling polynomials Sk(x) can be computed by the closed form k " k ` # X X s(` + 1; m + 1) X k + ` S (x) = (−1)kk! (−1)i S(k + i; i) xm: (2) k (k + `)! ` − i m=0 `=m i=0 2. Proof of Theorem 1 In combinatorics, the Bell polynomials of the second kind, or say, partial Bell polynomials, denoted by Bn;k(x1; x2; : : : ; xn−k+1) for n ≥ k ≥ 0, are defined by n−k+1 ` X n! Y xi i B (x ; x ; : : : ; x ) = : n;k 1 2 n−k+1 Qn−k+1 `i! i! 1≤i≤n;`i2f0g[N i=1 i=1 Pn i=1 i`i=n Pn i=1 `i=k See [1, p. 134, Theorem A]. They satisfy 2 n−k+1 k n Bn;k abx1; ab x2; : : : ; ab xn−k+1 = a b Bn;k(x1; x2; : : : ; xn−k+1) (3) for complex numbers a and b, see [1, p. 135], and k 1 1 1 n! X n + k B ; ;:::; = (−1)k−i S(n + i; i); (4) n;k 2 3 n − k + 2 (n + k)! k − i i=0 see [3, Theorem 1 and Remark 1], [4, Remark 2.1], [5, p. 30], [8, p. 315], [10, Lemma 2.3], and [13, Example 4.2]. The Fa`adi Bruno formula can be described in terms of the Bell polynomials of the second kind Bn;k(x1; x2; : : : ; xn−k+1) by n dn X f ◦ h(t) = f (k)(h(t))B h0(t); h00(t); : : : ; h(n−k+1)(t): (5) d tn n;k k=0 See [1, p. 139, Theorem C]. −(x+1) 1−e−t Taking f(u) = u and u = h(t) = t in the formula (5) and using the limit 1 − e−t lim u = lim h(t) = lim = 1 t!0 t!0 t!0 t Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 March 2017 doi:10.20944/preprints201703.0055.v1 Peer-reviewed version available at Tbilisi Mathematical Journal 2017, 10, 153-158; doi:10.1515/tmj-2017-0053 A CLOSED FORM FOR STIRLING POLYNOMIALS 3 yield k x+1 k d t X h−(x + 1)i` = B h0(t); h00(t); : : : ; h(k−`+1)(t) d tk 1 − e−t u(x+1)+` k;` `=0 k X 0 00 (k−`+1) ! h−(x + 1)i`Bk;` lim h (0); lim h (0);:::; lim h (0) t!0 t!0 t!0 `=0 as t ! 0, where n−1 ( Y x(x − 1) ··· (x − n + 1); n ≥ 1 hxin = (x − `) = 1; n = 0 `=0 is the falling factorial of x 2 R for n 2 f0g [ N. It is not difficult to see that `−1 `−1 Y ` Y h−(x + 1)i` = [−(x + 1) − m] = (−1) (x + m + 1) m=0 m=0 ` ` Y (−1)` Y (−1)` = (−1)` (x + m) = (x + m) = (x) ; x x `+1 m=1 m=0 where (x)n is defined by (1). Hence, it follows that `+1 (−1)` X h−(x + 1)i = s(` + 1; m)xm; ` ≥ 0: ` x m=0 Since Z 1 (−1)` h(`)(t) = st−1(ln s)` d s ! ; t ! 0; ` ≥ 0; 1=e 1 + ` by virtue of (3) and (4), we have k−`+1 0 00 (k−`+1) 1 1 (−1) Bk;` lim h (0); lim h (0);:::; lim h (0) = Bk;` − ; ;:::; t!0 t!0 t!0 2 3 k − ` + 2 ` 1 1 1 (−1)k+`k! X k + ` = (−1)kB ; ;:::; = (−1)i S(k + i; i): k;` 2 3 k − ` + 2 (k + `)! ` − i i=0 Accordingly, we obtain dk t x+1 lim t!0 d tk 1 − e−t k " `+1 #" ` # X (−1)` X (−1)k+`k! X k + ` = s(` + 1; m)xm (−1)i S(k + i; i) x (k + `)! ` − i `=0 m=0 i=0 k " ` #" `+1 # (−1)kk! X 1 X k + ` X = (−1)i S(k + i; i) s(` + 1; m)xm x (k + `)! ` − i `=0 i=0 m=0 k+1 " k ` # (−1)kk! X X s(` + 1; m) X k + ` = (−1)i S(k + i; i) xm x (k + `)! ` − i m=0 `=m−1 i=0 The explicit formula (2) is thus proved. The proof of Theorem 1 is complete. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 March 2017 doi:10.20944/preprints201703.0055.v1 Peer-reviewed version available at Tbilisi Mathematical Journal 2017, 10, 153-158; doi:10.1515/tmj-2017-0053 4 F. QI AND B.-N. GUO References [1] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 1974. [2] B.-N. Guo, I. Mez}o,and F. Qi, An explicit formula for the Bernoulli polynomials in terms of the r-Stirling numbers of the second kind, Rocky Mountain J. Math. 46 (2016), no. 6, 1919{1923; Available online at http://dx.doi.org/10.1216/RMJ-2016-46-6-1919. [3] B.-N. Guo and F. Qi, An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind, J. Anal. Number Theory 3 (2015), no. 1, 27{30; Available online at http://dx.doi.org/10.12785/jant/030105. [4] B.-N. Guo, F. Qi, Explicit formulas for special values of the Bell polynomials of the second kind and the Euler numbers, ResearchGate Technical Report (2015), http://dx.doi.org/ 10.13140/2.1.3794.8808. [5] B.-N. Guo and F. Qi, On inequalities for the exponential and logarithmic functions and means, Malays. J. Math. Sci. 10 (2016), no. 1, 23{34. [6] B.-N. Guo and F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math. 255 (2014), 568{579; Available online at http://dx.doi. org/10.1016/j.cam.2013.06.020.