In the paper, the author presents diagonal recurrence relations for the
Stirling numbers of the first kind. As by-products, the author also recovers
three explicit formulas for special values of the Bell polynomials of the
second kind.Comment: 7 page

Qi, Feng

English

arXiv

In the paper, the author presents diagonal recurrence relations for the Stirling numbers of the first kind. As by-products, the author also recovers three explicit formulas for special values of the Bell polynomials of the second kind

Contributions to Discrete Mathematics (E-Journal, University of Calgary)

Volume 11, Number 1, Pages 22–30ISSN 1715-0868DIAGONAL RECURRENCE RELATIONS FOR THESTIRLING NUMBERS OF THE FIRST KINDFENG QIAbstract. The paper presents diagonal recurrence relations for theStirling numbers of the first kind and recovers three explicit formulasfor special values of the Bell polynomials of the second kind.1. IntroductionIn combinatorics, the Bell polynomials of the second kind, also known asthe partial Bell polynomials, denoted by Bn,k(x1, x2, . . . , xn−k+1) for n ≥k ≥ 0, are defined by(1.1) Bn,k(x1, x2, . . . , xn−k+1) =∑1≤i≤n,`i∈{0}∪N∑ni=1 i `i=n∑ni=1 `i=kn!∏n−k+1i=1 `i!n−k+1∏i=1(xii!)`i;see [1, p. 134, Theorem A]. For more information on the Bell polynomialsin general and Dyck paths in particular, please see the papers [7, 8] and thereferences therein.In mathematics, the Stirling numbers arise in a variety of combinatoricsproblems and were introduced by James Stirling in the eighteenth century.There are two different kinds of the Stirling numbers. The Stirling numbersof the first kind, s(n, k), which are also called the signed Stirling numbersof the first kind, can be generated by(1.2)[ln(1 + x)]kk!=∞∑n=ks(n, k)xnn!, |x| < 1.The unsigned Stirling numbers of the first kind, (−1)n−ks(n, k), can beinterpreted as the number of permutations of {1, 2, . . . , n} with k cycles.Received by the editors December 10, 2014, and in revised form June 29, 2015.2010 Mathematics Subject Classification. Primary: 11B73; Secondary: 11B37, 11Y55,33B10.Key words and phrases. Stirling number of the first kind; diagonal recurrence relation;integral representation; Bell polynomial of the second kind; Faa` di Bruno’s formula; Lahnumber.The author was partially supported by the National Natural Science Foundation ofChina under Grant No. 11361038.c©2016 University of Calgary22DIAGONAL RECURRENCE RELATIONS FOR STIRLING NUMBERS 23Several “triangular,” “horizontal,” and “vertical” recurrence relations forthe Stirling numbers of the first kind s(n, k) are listed in [1, pp. 214–215,Theorems A, B, and C] ass(n, k) = s(n− 1, k − 1)− (n− 1)s(n− 1, k),(n− k)s(n, k) =∑k+1≤`≤n(−1)`−k(`k − 1)s(n, `),s(n, k) =∑k≤`≤ns(n + 1, ` + 1)n`−k,ks(n, k) =∑k−1≤`≤n−1(−1)n−`−1(n`)s(`, k − 1),s(n + 1, k + 1) =∑k≤`≤n(−1)`−1n−∏`q=1(` + q)s(`, k),where, by convention, the empty product equals 1. Observe that the term(−1)`−1 in the last recurrence relation was misprinted as (−1)n−1 in [1,p. 215, Theorem B].The aim of this paper is to present diagonal recurrence relations for theStirling numbers of the first kind s(n, k) based on an integral representation,Faa` di Bruno’s formula, and properties of the Bell polynomials of the secondkind Bn,k. Three explicit formulas for special values of the Bell polynomialsof the second kind Bn,k are recovered as by-products.The main results are formulated in the following theorem.Theorem 1.1. For n ≥ k ≥ 1, we haveBn,k(1!2,2!3, . . . ,(n− k + 1)!n− k + 2)= (−1)n−k 1k!k∑m=1(−1)m(km)(n+mn)s(n + m,m),(1.3)Bn,k(0, 1!, . . . ,(n− k)!)= (−1)n−k(nk) k∑m=0(−1)m(km)(n−mn−k)s(n−m, k −m),(1.4)ands(n, k) = (−1)kn∑m=1(−1)mk−1∑`=k−m(−1)`(n`)(`k −m)s(n− `, k − `)(1.5)= (−1)n−kk−1∑`=0(−1)`(n`)(`− 1k − n− 1)s(n− `, k − `),(1.6)24 FENG QIwhere the conventions that(00)= 1,(−1−1)= 1, and(pq)= 0 for p ≥ 0 > q areadopted in Equation (1.6).2. Proof of Theorem 1.1Recently, three integral representations for the unsigned Stirling numbersof the first kind (−1)n−ks(n, k) were discovered in [10]. The first amongthem, [10, Theorem 2.1], states that, for 1 ≤ k ≤ n,(2.1) s(n, k) =(nk)limx→0dn−kdxn−k{[∫ ∞0(∫ 11/etxu−1 dt)e−u du]k}.In combinatorial analysis, Faa` di Bruno’s formula plays an important roleand may be described in terms of the Bell polynomials of the second kindBn,k by(2.2)dndtn[f ◦ h(t)] =n∑k=1f (k)(h(t))Bn,k(h′(t), h′′(t), . . . , h(n−k+1)(t));see, for instance, [1, p. 139, Theorem C]. The Bell polynomials of the secondkind, Bn,k, satisfy∞∑n=kBn,k(x1, x2, . . . , xn−k+1)tnn!=1k!( ∞∑m=1xmtmm!)k,(2.3)Bn,k(abx1, ab2x2, . . . ,abn−k+1xn−k+1)= akbnBn,k(x1, xn, . . . , xn−k+1),(2.4)Bn,k(x22,x33, . . . ,xn−k+2n− k + 2)=n!(n + k)!Bn+k,k(0, x2, . . . , xn+1),(2.5)where a and b are arbitrary complex numbers; see [1, pp. 133, 135–136] fordetails.Let(2.6) h(x) =∫ ∞0(∫ 11/etxu−1 dt)e−u du.It is clear that, for ` ∈ N,h(`)(x) =∫ ∞0[∫ 11/etxu−1(ln t)` dt]u`e−u du→∫ ∞0[∫ 11/e(ln t)`tdt]u`e−u du =(−1)``!` + 1as x → 0. Setting f(v) = vk in Equation (2.2) and using the functiondescribed in (2.6) to compute (2.1), we obtains(n, k) =(nk)limx→0n−k∑m=1f (m)(h(x))Bn−k,m(h′(x), . . . , h(n−k−m+1)(x))(2.7)DIAGONAL RECURRENCE RELATIONS FOR STIRLING NUMBERS 25=(nk)limx→0k∑m=1f (m)(h(x))Bn−k,m(h′(x), . . . , h(n−k−m+1)(x)),n > 2k;(nk)limx→0n−k∑m=1f (m)(h(x))Bn−k,m(h′(x), . . . , h(n−k−m+1)(x)),k ≤ n ≤ 2k=(nk)limx→0k∑m=1f (m)(h(0))Bn−k,m(h′(0), . . . , h(n−k−m+1)(0)),n > 2k;(nk)limx→0n−k∑m=1f (m)(h(0))Bn−k,m(h′(0), . . . , h(n−k−m+1)(0)),k ≤ n ≤ 2k=(nk) k∑m=1k!(k −m)!Bn−k,m(− 1!2, . . .. . . ,(−1)n−k−m+1(n− k −m + 1)!n− k −m + 2), n > 2k;(nk) n−k∑m=1k!(k −m)!Bn−k,m(− 1!2, . . .. . . ,(−1)n−k−m+1(n− k −m + 1)!n− k −m + 2), k ≤ n ≤ 2kTaking xm = m!/(m + 1) in Equation (2.3) and using (1.2) give∞∑n=kBn,k(1!2,2!3, . . . ,(n− k + 1)!n− k + 2)tnn!=1k!( ∞∑m=1tmm + 1)k=(−1)kk![ln(1− t)t+ 1]k=(−1)kk!k∑i=0(ki)[ln(1− t)t]i= (−1)k 1k!k∑i=0(ki)i!ti∞∑`=i(−1)`s(`, i) t``!= (−1)kk∑i=01(k − i)!∞∑`=i(−1)`s(`, i) t`−i`!.This implies thatBn,k(1!2,2!3, . . . ,(n− k + 1)!n− k + 2)= n!(−1)kk∑i=0(−1)n+i(k − i)!s(n + i, i)(n + i)!26 FENG QI= (−1)n−k 1k!k∑i=0(ki)(n+ii)(−1)is(n + i, i),from here Equation (1.3) follows.Substituting Equation (1.3) into (2.5) leads toBn+k,k(0, 1!, 2! . . . , n!) = (−1)n−k(n + kk) k∑i=0(−1)i(ki)(n+ii)s(n + i, i),which may be rewritten as Equation (1.4).By virtue of Equation (2.4), we have(2.8) Bn−k,m(−12,23, . . . ,(−1)n−k−m+1(n− k −m + 1)!n− k −m + 2)= (−1)n−kBn−k,m(1!2,2!3, . . . ,(n− k −m + 1)!n− k −m + 2).After substituting Equation (1.3) into (2.8), then into (2.7), and finallysimplifying, we find that when 2k ≥ n ≥ k ≥ 1,(2.9) s(n, k) =n−k∑m=1m∑`=1(−1)m+`(nk − `)(k − `m− `)s(n− k + `, `),and when n > 2k > 0,(2.10) s(n, k) =k∑m=1m∑`=1(−1)m+`(nk − `)(k − `m− `)s(n− k + `, `).Using the convention that s(n, k) = 0 for 0 ≤ n < k, we can unify Equations(2.9) and (2.10) into(2.11) s(n, k) =n∑m=1m∑`=1(−1)m+`(nk − `)(k − `k −m)s(n− k + `, `),which can be further formulated as Equation (1.5).Interchanging the two sums in (1.5) and computing the inner sum yieldss(n, k) = (−1)kk−1∑`=k−n(−1)`(n`)[ n∑m=k−`(−1)m(`k −m)]s(n− `, k − `)= (−1)n−kk−1∑`=k−n(−1)`(n`)(`− 1k − n− 1)s(n− `, k − `),which can be rewritten as Equation (1.6). The proof of Theorem 1.1 iscomplete.DIAGONAL RECURRENCE RELATIONS FOR STIRLING NUMBERS 273. RemarksRemark 3.1. The recurrence relations (1.5) and (1.6) are neither “trian-gular”, nor “vertical”, nor “horizontal” recurrence relations as listed in [1,pp. 214–215, Theorems A, B, and C], so we call them “diagonal” recurrencerelations for the Stirling numbers of the first kind s(n, k).Remark 3.2. Equation (1.5) is also true if we change the sum over m from1 to k instead of from 1 to n.Remark 3.3. From [11, Corollary 2.3], we may compute the Stirling num-bers of the first kind, s(n, k), for 2 ≤ k ≤ n by the equations(n, k) = (−1)n−k(n− 1)!n−1∑`1=k−11`1`1−1∑`2=k−21`2· · ·`k−3−1∑`k−2=21`k−2`k−2−1∑`k−1=11`k−1.This equation may be reformulated as(−1)n−k s(n, k)(n− 1)! =n−1∑m=k−11m[(−1)m−(k−1) s(m, k − 1)(m− 1)!].Remark 3.4. By applying the integral representation described in Equa-tion (2.1), some properties for the Stirling numbers of the first kind s(n, k),including the logarithmic convexity with respect to n ≥ 0 of the sequence{|s(n + k, k)|(n+kk) }n≥0for any fixed k ∈ N (see [10, Corollary 5.1]), were established in [10, Sec-tion 5].Remark 3.5. It is well known in combinatorics that(3.1) Bn,k(1!, 2!, . . . , (n− k + 1)!) =(nk)(n− 1k − 1)(n− k)!for n ≥ k ≥ 1; see [1, p. 135, Theorem B]. We now recover this identity inan alternative way.In [14, Theorems 2.1 and 2.2], it was inductively obtained, for i ∈ N andt 6= 0,(3.2)die1/tdti= (−1)ie1/t 1t2ii−1∑k=0ai,ktkand(3.3)die−1/tdti=e−1/tt2ii−1∑k=0(−1)kai,ktk,whereai,k =(ik)(i− 1k)k!28 FENG QIfor all 0 ≤ k ≤ i − 1 and an,n−k are the Lah numbers L(n, k); see also [13,Equations (1.3) and (1.4)]. For more information on the Lah numbersL(n, k), please refer to the recent references [2, 5, 6] and related referencestherein.From Equations (2.2) and (2.4), it follows that, for i ∈ N and t 6= 0,die1/tdti= e1/ti∑k=1Bi,k(−1!t2,2!t3, . . . , (−1)i−k+1 (i− k + 1)!ti−k+2)= (−1)ie1/ti∑k=11ti+kBi,k(1!, 2!, . . . , (i− k + 1)!)(3.4)anddie−1/tdti= e−1/ti∑k=1Bi,k(1!t2,−2!t3, . . . , (−1)i−k (i− k + 1)!ti−k+2)= e−1/ti∑k=1(−1)kBi,k(−1!t2,2!t3, . . . , (−1)i−k+1 (i− k + 1)!ti−k+2)= e−1/ti∑k=1(−1)i+kti+kBi,k(1!, 2!, . . . , (i− k + 1)!).Combining Equation (3.2) with (3.4) and Equation (3.3) with the aboveequation, respectively, shows(−1)i 1t2ii−1∑k=0ai,ktk = (−1)ii∑k=11ti+kBi,k(1!, 2!, . . . , (i− k + 1)!)and1t2ii−1∑k=0(−1)kai,ktk =i∑k=1(−1)i+kti+kBi,k(1!, 2!, . . . , (i− k + 1)!).As a result,i∑k=1ai,i−ktk =i∑k=1Bi,k(1!, 2!, . . . , (i− k + 1)!)tk,which impliesBn,k(1!, 2!, . . . , (n− k + 1)!) = an,n−k= (n− k)!(nn− k)(n− 1n− k)= (n− k)!(nk)(n− 1k − 1).Thus we have reestablished the identity described in Equation (3.1).Remark 3.6. We can find Equation (1.3) in [15].DIAGONAL RECURRENCE RELATIONS FOR STIRLING NUMBERS 29Remark 3.7. In [3, 4, 12, 15] and the related references therein, severalspecial values of the Bell polynomials of the second kind Bn,k are discovered,collected, and applied.Remark 3.8. This paper is a revised version of the preprint [9].AcknowledgementsThe author thanks the anonymous referees and technical editors for theircareful corrections to and valuable comments on the original version of thispaper.References1. L. Comtet, Advanced combinatorics: The art of finite and infinite expansions, revisedand enlarged edition ed., D. Reidel Publishing Co., 1974.2. S. Daboul, J. Mangaldan, M. Z. Spivey, and P. J. Taylor, The Lah numbers andthe nth derivative of e1/x, Math. Mag. 86 (2013), no. 1, 39–47; available online athttp://dx.doi.org/10.4169/math.mag.86.1.039.3. B.-N. Guo and F. Qi, Explicit formulas for special values of the Bell polynomials of thesecond kind and the Euler numbers, ResearchGate Technical Report (2015); availableonline at http://dx.doi.org/10.13140/2.1.3794.8808.4. , An explicit formula for Bernoulli numbers in terms of Stirling numbers ofthe second kind, J. Anal. Number Theory 3 (2015), no. 1, 27–30; available online athttp://dx.doi.org/10.12785/jant/030105.5. , Six proofs for an identity of the Lah numbers, Online J. Anal. Comb. 10(2015), 5 pp.6. J. Lindsay, T. Mansour, and M. Shattuck, A new combinatorial interpretation of aq-analogue of the Lah numbers, J. Comb 2 (2011), no. 2, 245–264; available online athttp://dx.doi.org/10.4310/JOC.2011.v2.n2.a4.7. T. Mansour and Y. Sun, Bell polynomials and k-generalized Dyck paths, DiscreteAppl. Math. 156 (2008), no. 12, 2279–2292; available online at http://dx.doi.org/10.1016/j.dam.2007.10.009.8. , Dyck paths and partial Bell polynomials, Australas. J. Combin. 42 (2008),285–297.9. F. Qi, A recurrence formula for the first kind Stirling numbers, arXiv preprint (2013);available online at http://arxiv.org/abs/1310.5920.10. , Integral representations and properties of Stirling numbers of the first kind,J. Number Theory 133 (2013), no. 7, 2307–2319; available online at http://dx.doi.org/10.1016/j.jnt.2012.12.015.11. , Explicit formulas for computing Bernoulli numbers of the second kind andStirling numbers of the first kind, Filomat 28 (2014), no. 2, 319–327; available onlineat http://dx.doi.org/10.2298/FIL1402319O.12. , Derivatives of tangent function and tangent numbers, Appl. Math. Comput.268 (2015), 844–858; available online at http://dx.doi.org/10.1016/j.amc.2015.06.123.13. F. Qi and C. Berg, Complete monotonicity of a difference between the exponen-tial and trigamma functions and properties related to a modified Bessel function,Mediterr. J. Math. 10 (2013), no. 4, 1685–1696, available online at http://dx.doi.org/10.1007/s00009-013-0272-2.30 FENG QI14. X.-J. Zhang, F. Qi, and W.-H. Li, Properties of three functions relating to the ex-ponential function and existence of partitions of unity, Int. J. Open Probl. Com-put. Sci. Math. 5 (2012), no. 3, 122–127; available online at http://dx.doi.org/10.12816/0006128.15. Z.-Z. Zhang and J.-Z. Yang, Notes on some identities related to the partial Bell poly-nomials, Tamsui Oxf. J. Inf. Math. Sci 28 (2012), no. 1, 39–48.Institute of Mathematics, Henan Polytechnic University, Jiaozuo City,Henan Province, 454010, China;College of Mathematics, Inner Mongolia University for Nationalities,Tongliao City, Inner Mongolia Autonomous Region, 028043, China;Department of Mathematics, College of Science, Tianjin PolytechnicUniversity, Tianjin City, 300387, ChinaE-mail address: qifeng618@gmail.com, qifeng618@hotmail.comURL: https://qifeng618.wordpress.com

Diagonal recurrence relations for the Stirling numbers of the first kind

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Diagonal recurrence relations for the Stirling numbers of the first kind

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