arXiv:1407.7429v1 [math.CO] 28 Jul 2014 offiinspae oemil nteter fcomposition of t theory Since the [9]. in c Euler mainly by role i addressed a time were played first also coefficients they the later for and appeared 41] presumably p. they form, written In aiu aesadfo ieetprpcie 1 ,3 ,7 8 7, 4, 3, 2, [1, modificati perspectives their different from and and coefficients papers various binomial extended the Thus, .7],aedfie stececet nteexpansion the in coefficients the as defined pol are called 77]), occasionally p. coefficients, binomial extended The Introduction 1 ahmtc ujc lsicto (2010) Classification Subject Mathematics Hermit approximation, ber normal theorem, limit central local Keywords exp coefficients all state numbers. we Bernoulli and and theorem polynomials limit We central terms. local error a uniform with expansion asymptotic complete Abstract ( ∗ oeo xeddBnma Coefficients Binomial Extended on Note A ,n q n, k, eateto ahmtc,K evn eetjela 200 Celestijnenlaan Leuven, KU Mathematics, of Department -al [email protected] E-mail: fcmoiin of compositions of ) esuytedsrbto fteetne ioilcoefficien binomial extended the of distribution the study We X k ∞ xeddbnma offiin,cmoiin opeeasymp complete composition, coefficient, binomial extended =0  n k  ( q ) x k = k + 1 with c hrtnNeuschel Thorsten ( x ,n q n, k, + n x at o exceeding not parts 2 = ) + · · ·  1 k + − rmr 18;Scnay0A6 41A60. 05A16, Secondary 11P82; Primary n x n q
 n ( q ,q n, , − o 40 E30 evn Belgium. Leuven, BE-3001 2400, box B 1) . ∗ q oyoil enul num- Bernoulli polynomial, e nma offiins(.. [5, (e.g., coefficients ynomial ∈ sgvnby given is 0 2 3 5,adamong and 15], 13, 12, 10, , e,teetne binomial extended the hen, fitgr stenumber the as integers of s iil ssm fHermite of sums as licitly banteepninfrom expansion the obtain N ok yD ove[6, Moivre De by works n n aebe tde in studied been have ons = { 1 , 2 . . . , oi expansion, totic sb eiiga deriving by ts } . (1.1) the properties their distribution is of particular interest. Recently, Eger [8] showed (using a slightly different notation) that

n (q) (q + 1)n , nq/2 ∼ q(q+2)   2πn 12 q 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 1.1. For all integers N 2 we have ≥ (q) [(N−2)/2] q(q + 2)n 1 n 1 − 2 q (x) 1 = e x /2 + 2ν + o , 12 (1 + q)n k √ nν n(N−2)/2 r 2π ν=1   X   as n , uniformly with respect to all k Z, with →∞ ∈ √12 q x = k n , q(q + 2)n − 2   p where the functions q2ν (x) are given explicitly as sums of Hermite polynomials and Bernoulli numbers (see Theorem 2.2 below for the exact formulae). Although we only deal with the very basic situation of the extended binomial coefficients in (1.1) 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.

2 Proof of the main result

First of all we fix some notations following Petrov [14]. For a (real) random variable X we denote its characteristic function by

ϕ (t)= EeitX , t R, X ∈ 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 ϕX is k times continuously differentiable on R and we have

k d 1 k ϕX (t) = EX . dtk t=0 ik

2 Moreover, in this case we define the cumulants of order k by 1 dk γk = log ϕX (t) , ik dtk t=0

where the logarithm takes its principal branch. Now, let (Xn) be a sequence of indepen- dent integer-valued random variables having a common distribution and suppose that for all positive integer values of k we have E X k < | 1| ∞ and 2 EX1 = µ, V arX1 = σ > 0. Thus, for the sum given by n Sn = Xν ν=1 X we obtain 2 ESn = nµ, V arSn = nσ , and for integer k we define the probabilities

pn(k)= P (Sn = k) . Furthermore, we introduce the Hermite polynomials (in the probabilist’s version) m 2 d − 2 H (x) = ( 1)mex /2 e x /2, m − dxm and for positive integers ν we define the functions

ν km 1 −x2/2 1 γm+2 qν(x)= e Hν+2s(x) m+2 , (2.1) √2π km! (m + 2)!σ k1,...,kν ≥0 m=1   k1+2k2X+···+νkν =ν Y where s = k + + k and γ denotes the cumulant of order m +2 of X . 1 · · · ν m+2 1 Finally, we demand (for convenience) that the maximal span of the distribution of X1 is equal to one. This means that there are no numbers a and h> 1 such that the values taken on by X with probability one can be expressed in the form a+hk (k Z). Under 1 ∈ all these assumptions we have the following complete asymptotic expansion in the sense of a local central limit theorem [14, p. 205]. Theorem 2.1. For all integers N 2 we have ≥ N−2 1 −x2/2 qν(x) 1 σ√npn(k)= e + + o , (2.2) √ nν/2 n(N−2)/2 2π ν=1 X   as n , uniformly with respect to all k Z, where we have →∞ ∈ k nµ x = − . σ√n

3 In the following we choose X to take the integer values 0,...,q with 1 { } 1 P (X = k)= , k 0,...,q . 1 q + 1 ∈ { } Hence, we obtain

1 n (q) p (k)= P (S = k)= , k Z. (2.3) n n (1 + q)n k ∈   It is our aim to apply Theorem 2.1 in full generality and we want compute all cumulants as explicit as possible.

Lemma 2.1. For the k-th order cumulant γk of X1 we have

q 2 , if k = 1; γk = 0, if k odd and k > 1; (2.4) B  2l (q + 1)2l 1 , if k = 2l, l 1, 2l − ≥ where , ν 0, denote the Bernoulli numbers,
e.g., [11, p. 22]. Bν ≥ Proof. First, we observe that the characteristic function of X1 is given by 1+ eit + + eqit ϕ (t)= · · · . X1 1+ q According to the definition of the cumulants we obtain for a positive integer k 1 dk γk = log ϕX1 (t) ik dtk t=0 k 1 d it qit = log 1+ e + + e log(1 + q) ik dtk · · · − t=0

1 dk  e(q+1)it 1

= k k log it − i dt e 1 ! t=0 − k q+1 1 d q sin 2 t = k k it + log t i dt (2 sin !) t=0 2 k q+1 t q 1 d sin 2 t sin 2 = δk,1 + k k log q+1 log t , 2 i dt ( t ! − ) t=0 2  2  where δk,1 denotes the Kronecker delta. Using d sin z 1 log = cotan z dz z − z   yields q 1 dk−1 q + 1 q + 1 2 1 t 2 γk = δk,1 + − cotan t cotan . 2 ik dtk 1 2 2 − (q + 1)t − 2 2 − t t=0     

4 Now, making use of the following expansion (see, e.g., [11, p. 35])

∞ m 1 4 − cotan z = ( 1)m z2m 1 , 0 < z < π, − z − (2m)!B2m | | mX=1 after some algebra we obtain

∞ k−1 q 1 d m 2m 2m 2m−1 γk = δk,1 + − ( 1) B (q + 1) 1 t . 2 ik dtk 1 − (2m)! − t=0 m=1 X
Carrying out the differentiation under the summation sign immediately gives us (2.4).

Remark 2.1. As an immediate consequence of Lemma 2.1 we obtain q EX = µ = γ = 1 1 2 and, as we know = 1 , B2 6 q(q + 2) V arX = σ2 = γ = B2 (q + 1)2 1 = . 1 2 2 − 12
We now are ready to state the main theorem in form of a complete asymptotic expan- n (q) sion with explicit coefficients for the extended binomial coefficients k . Theorem 2.2. For all integers N 2 we have
≥ (q) [(N−2)/2] q(q + 2)n 1 n 1 − 2 q (x) 1 = e x /2 + 2ν + o , 12 (1 + q)n k √ nν n(N−2)/2 r 2π ν=1   X   as n , uniformly with respect to all k Z, with →∞ ∈ √12 q x = k n , q(q + 2)n − 2   and p ν 1 12 −x2/2 q2ν(x)= e (2.5) √2π q(q + 2)   s ν 2m+2 k2m 6 1 2(m+1) (q + 1) 1 H2(ν+s)(x) B − , × q(q + 2) k2m! (2m + 2)!(m + 1) k2,k4,...,k2ν ≥0   m=1
! k2+2k4+X···+νk2ν =ν Y where s = k + k + + k . 2 4 · · · 2ν

5 Proof. The proof is based on an application of Theorem 2.1 to the probabilities defined in (2.3). First we observe that in our situation the functions given in (2.1) vanish identically for odd indices, which turns out to be a consequence of (2.4). Indeed, if ν = 2l +1 for an integer l 0, then in every solution k ,...,k 0 of the equation ≥ 1 2l+1 ≥ k + 2k + + (2l + 1)k = 2l + 1 1 2 · · · 2l+1 there is at least one odd index i with ki > 0. Consequently, using (2.4) we have

2l+1 1 γ km m+2 = 0, k ! (m + 2)!σm+2 m=1 m Y   from which follows that q2l+1(x) vanishes identically. Thus, only the functions q2ν(x) appear in (2.2) and here we have

2ν km 1 −x2/2 1 γm+2 q2ν(x)= e H2(ν+s)(x) m+2 , √2π km! (m + 2)!σ k1,...,k2ν ≥0 m=1   k1+2k2+X···+2νk2ν =2ν Y where s = k + + k . An analogous argument as in the odd case above shows that 1 · · · 2ν a solution k1,...,k2ν of the equation k + 2k + + 2νk = 2ν 1 2 · · · 2ν with a positive entry at an odd index does not give any contribution to the whole sum, so that we can write

ν k2m 1 −x2/2 1 γ2m+2 q2ν(x)= e H2(ν+s)(x) 2m+2 , √2π k2m! (2m + 2)!σ k2,k4,...,k2ν ≥0 m=1   k2+2k4+X···+νk2ν =ν Y where s = k + k + + k . Now, taking the explicit form of the cumulants in (2.4) 2 4 · · · 2ν into account, after some elementary computation we obtain (2.5).

Remark 2.2. As a concluding remark we state the meaning of Theorem 2.2 for N = 5 explicitly. Using the known facts 1 H (x)= x4 6x2 + 3, = , 4 − B4 −30 we obtain (q) 4 4 2 q(q + 2)n 1 n 1 − 2 (q + 1) 1 x 6x + 3 1 = e x /2 1 − − +o , 12 (1 + q)n k √ − 20nq2(q + 2)2 n3/2 r   2π (

)   as n , uniformly with respect to all k Z, where we have →∞ ∈ √12 q x = k n . q(q + 2)n − 2   p

6 3 Acknowledgments

This work is supported by KU Leuven research grant OT/12/073 and the Belgian In- teruniversity Attraction Pole P07/18.

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