A Note on Extended Binomial Coefficients

Transcription

1 A Note on Extended Binomial Coefficients Thorsten Neuschel arxiv: v [math.co] 8 Jul 04 Abstract We study the distribution of the extended binomial coefficients by deriving a complete asymptotic expansion with uniform error terms. We obtain the expansion from a local central limit theorem and we state all coefficients explicitly as sums of Hermite polynomials and Bernoulli numbers. Keywords extended binomial coefficient, composition, complete asymptotic expansion, local central limit theorem, normal approximation, Hermite polynomial, Bernoulli number Mathematics Subject Classification 00) Primary P8; Secondary 05A6, 4A60. Introduction The extended binomial coefficients, occasionally called polynomial coefficients e.g., [5, p. 77]), are defined as the coefficients in the expansion k=0 ) n q) x k = +x+x + +x q) n, n,q N = {,,...}..) k In written form, they presumably appeared for the first time in works by De Moivre [6, p. 4] and later they also were addressed by Euler [9]. Since then, the extended binomial coefficients played a role mainly in the theory of compositions of integers as the number ck,n,q) of compositions of k with n parts not exceeding q is given by ck,n,q) = ) n q ). k n Thus, the extended binomial coefficients and their modifications have been studied in various papers and from different perspectives [,, 3, 4, 7, 8, 0,, 3, 5], and among Department of Mathematics, KU Leuven, Celestijnenlaan 00B box 400, BE-300 Leuven, Belgium. Thorsten.Neuschel@wis.kuleuven.be

2 the properties their distribution is of particular interest. Recently, Eger[8] showed using a slightly different notation) that ) n q) nq/ q +)n πn qq+) as n, meaning that the quotient of both sides tends to unity. Moreover, based upon numerical simulations [8] the question arises how well those coefficients can be approximated by normal approximations in general. It is the aim of this note to give a precise and comprehensive answer to this question by establishing a complete asymptotic expansion for the extended binomial coefficients with error terms holding uniformly with respect to all integer k. More precisely, we show the following. Theorem.. For all integers N we have qq +)n +q) n ) n q) = [N )/] e x / q ν x) + k π n ν +o ν= as n, uniformly with respect to all k Z, with x = k q ) qq +)n n,, n N )/ ), where the functions q ν x) are given explicitly as sums of Hermite polynomials and Bernoulli numbers see Theorem. below for the exact formulae). Although we only deal with the very basic situation of the extended binomial coefficients in.) here, the presented approach is a general one, which admits the derivation of complete) asymptotic expansions in many applications. However, it is not always possible to obtain the involved quantities in a very explicit form, which is an instance making the case of the extended binomial coefficients further interesting and worth to be presented. Proof of the main result First of all we fix some notations following Petrov [4]. For a real) random variable X we denote its characteristic function by ϕ X t) = Ee itx, t R, where, as usual, E means the mathematical expectation with respect to the underlying probability distribution. If X has finite moments up to k-th order, then ϕ X is k times continuously differentiable on R and we have d k dt kϕ Xt) = t=0 i kexk.

3 Moreover, in this case we define the cumulants of order k by γ k = d k i k dt k logϕ Xt), t=0 where the logarithm takes its principal branch. Now, let X n ) be a sequence of independent integer-valued random variables having a common distribution and suppose that for all positive integer values of k we have and Thus, for the sum given by we obtain E X k < EX = µ, VarX = σ > 0. S n = n ν= X ν ES n = nµ, VarS n = nσ, and for integer k we define the probabilities p n k) = P S n = k). Furthermore, we introduce the Hermite polynomials in the probabilist s version) H m x) = ) m e x / dm dx me x /, and for positive integers ν we define the functions q ν x) = ν e x / H ν+s x) π k,...,kν 0 k +k + +νkν=ν m= ) γ km m+ k m! m+)!σ m+,.) where s = k + +k ν and γ m+ denotes the cumulant of order m+ of X. Finally, we demand for convenience) that the maximal span of the distribution of X is equal to one. This means that there are no numbers a and h > such that the values taken on by X with probability one can be expressed in the form a+hk k Z). Under all these assumptions we have the following complete asymptotic expansion in the sense of a local central limit theorem [4, p. 05]. Theorem.. For all integers N we have σ np n k) = N e x / + π ν= q ν x) +o nν/ as n, uniformly with respect to all k Z, where we have x = k nµ σ n. n N )/ ),.) 3

4 In the following we choose X to take the integer values {0,...,q} with Hence, we obtain PX = k) = q +, k {0,...,q}. p n k) = PS n = k) = ) n q) +q) n, k Z..3) k It is our aim to apply Theorem. in full generality and we want compute all cumulants as explicit as possible. Lemma.. For the k-th order cumulant γ k of X we have q, if k = ; γ k = 0, if k odd and k > ; q +) l ), if k = l,l, B l l where B ν, ν 0, denote the Bernoulli numbers, e.g., [, p. ]. Proof. First, we observe that the characteristic function of X is given by ϕ X t) = +eit + +e qit. +q According to the definition of the cumulants we obtain for a positive integer k γ k = d k i k dt k logϕ X t) t=0 = d k { log +e it i k dt k + +e qit) log+q) } t=0 ) = d k i k dt k log e q+)it e it t=0 )} sin q+ = i k d k dt k { q it+log { = q δ k, + i k d k dt k t sin t t=0 sin q+ log t ) sin t ) } q+ t log t where δ k, denotes the Kronecker delta. Using ) d sinz dz log = cotanz z z yields γ k = q δ k, + i k d k dt k { q + t=0,.4) cotan q + ) t cotan t q +)t t)} t=0. 4

5 Now, making use of the following expansion see, e.g., [, p. 35]) cotanz z = ) m 4m m)! B mz m, 0 < z < π, m= after some algebra we obtain γ k = q δ k, + d k i k dt k m= ) m B m q +) m ) t m m)! t=0. Carrying out the differentiation under the summation sign immediately gives us.4). Remark.. As an immediate consequence of Lemma. we obtain and, as we know B = 6, VarX = σ = γ = B EX = µ = γ = q q +) ) = qq +). We now are ready to state the main theorem in form of a complete asymptotic expansion with explicit coefficients for the extended binomial coefficients n k) q). Theorem.. For all integers N we have qq +)n +q) n ) n q) = [N )/] e x / q ν x) + k π n ν +o ν= as n, uniformly with respect to all k Z, with x = k q ) qq +)n n, n N )/ ), and q ν x) = ) ν e x / π qq +) k,k 4,...,k ν 0 k +k 4 + +νk ν =ν H ν+s) x) 6 qq +) ) s ν m= k m!.5) Bm+) q +) m+ ) ) km, m+)!m+) where s = k +k 4 + +k ν. 5

6 Proof. The proof is based on an application of Theorem. to the probabilities defined in.3). First we observe that in our situation the functions given in.) vanish identically for odd indices, which turns out to be a consequence of.4). Indeed, if ν = l + for an integer l 0, then in every solution k,...,k l+ 0 of the equation k +k + +l +)k l+ = l+ there is at least one odd index i with k i > 0. Consequently, using.4) we have l+ m= ) γ km m+ k m! m+)!σ m+ = 0, from which follows that q l+ x) vanishes identically. Thus, only the functions q ν x) appear in.) and here we have q ν x) = π e x / k,...,k ν 0 k +k + +νk ν =ν H ν+s) x) ν m= ) γ km m+ k m! m+)!σ m+, where s = k + +k ν. An analogous argument as in the odd case above shows that a solution k,...,k ν of the equation k +k + +νk ν = ν with a positive entry at an odd index does not give any contribution to the whole sum, so that we can write q ν x) = ν ) e x / γ km m+ H ν+s) x) π k m! m+)!σ m+, k,k 4,...,k ν 0 k +k 4 + +νk ν =ν where s = k +k 4 + +k ν. Now, taking the explicit form of the cumulants in.4) into account, after some elementary computation we obtain.5). m= Remark.. As a concluding remark we state the meaning of Theorem. for N = 5 explicitly. Using the known facts we obtain qq +)n +q) n H 4 x) = x 4 6x +3, B 4 = 30, ) { n q) = e x / k π q +) 4 ) x 4 6x +3 ) } 0nq q +) as n, uniformly with respect to all k Z, where we have x = k q ) qq +)n n. +o ) n 3/, 6

7 3 Acknowledgments This work is supported by KU Leuven research grant OT//073 and the Belgian Interuniversity Attraction Pole P07/8. References [] G. Andrews, Euler s Exemplum Memorabile Inductionis Fallacis and q-trinomial Coefficients, J. Amer. Math. Soc ), [] N. Balakrishnan, K. Balasubramanian and R. Viveros, Some discrete distributions related to extended Pascal Triangles, Fibonacci Quart ) [3] C. Banderier and P. Hitczenko, Enumeration and asymptotics of restricted compositions having the same number of parts, Discrete Appl. Math. 60 8) 0) [4] C. Caiado and P. Rathie, Polynomial coefficients and distributions of the sum of discrete uniform variables, in A. Mathai, M. Pathan, K. Jose, J. Jacob, eds., Eighth Annual Conference of the Society of Special Functions and their Applications, Pala, India, Society for Special Functions and their Applications, 007. [5] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, 974. [6] A. De Moivre, The Doctrine of Chances: or, A Method of Calculating the Probabilities of Events in Play, 3rd ed. London: Millar, 756; rpt. New York: Chelsea, 967. [7] S. Eger, Restricted weighted integer compositions and extended binomial coefficients, Journal of Integer Sequences 6 03). [8] S. Eger, Stirling s approximation for central extended binomial coefficients, The American Mathematical Monthly 04), [9] L. Euler, De evolutione potestatis polynomialis cuiuscunque + x + x + x 3 + x 4 +etc.) n, Nova Acta Academiae Scientarum Imperialis Petropolitinae 80), [0] N. Fahssi, A systematic study of polynomial triangles, The Electronic Journal of Combinatorics 0). [] I. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products, twelfth printing, Academic Press, New York, 979. [] S. Heubach and T. Mansour, Compositions of n with parts in a set, Congressus Numerantium ). 7

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