On the Coefficients of the Asymptotic Expansion of n!

Gergő Nemes Loránd Eötvös University H-1117 Budapest Pázmány Péter sétány 1/C Hungary

                      nemesgery@gmail.com

Abstract Applying a theorem of Howard to a formula recently proved by Brassesco and Méndez, we derive new simple explicit formulas for the coefficients of the asymptotic expansion of the sequence of factorials.

1 Introduction

It is well known that the factorial of a positive integer n has the asymptotic expansion

n! ∼ n^{n}e^{−n}\sqrt{2πn} ∑_{k≥0} \frac{a_{k}}{n^{k}} ,  (1)

known as Stirling’s formula (see, e.g., [1, 3, 4]). The coefficients a_{k} in this series are usually called the Stirling coefficients [1, 6] (Sloane’s \underline{A001163} and \underline{A001164}) and can be computed from the sequence b_{k} defined by the recurrence relation

b_{k} = \frac{1}{k + 1} (b_{k−1} − ∑_{j=2}^{k−1} jb_{j}b_{k−j+1}) , b_{0} = b_{1} = 1,  (2)

since a_{k} = (2k + 1)!!b_{2k+1} [3, 4]. Here (2k + 1)!! = (2k + 1) · (2k − 1) ⋯ 5 · 3 · 1 is the double factorial. It was pointed out by Paris and Kaminski [6] that “There is no known closed-form representation for the Stirling coefficients”. However there is a closed-form expression that involves combinatorial quantities due to Comtet [5]:

a_{k} = ∑_{j=0}^{2k} (−1)^{j} \frac{d_{3} (2k + 2j, j)}{2^{k+j} (k + j)!},  (3)

where d_{3} (p, q) is the number of permutations of p with q permutation cycles all of which are ≥ 3 (Sloane’s \underline{A050211}). Brassesco and Méndez [7] proved in a recent paper that

a_{k} = ∑_{j=0}^{2k} (−1)^{j} \frac{S_{3} (2k + 2j, j)}{2^{k+j} (k + j)!},  (4)

where S_{3} (p, q) denotes the 3-associated Stirling numbers of the second kind (Sloane’s \underline{A059022}). We show that the Stirling coefficients a_{k} can be expressed in terms of the conventional Stirling numbers of the second kind (Sloane’s \underline{A008277}). A corollary of this result is an explicit, exact expression for the Stirling coefficients.

2 The formulas for coefficients

One of our main results is the following:

Theorem 1. The Stirling coefficients have a representation of the form

a_{k} = \frac{(2k)!}{2^{k}k!} ∑_{i=0}^{2k} \binom{k + i − 1/2}{i}\binom{3k + 1/2}{2k − i}2^{i} ∑_{j=0}^{i} \binom{i}{j} (−1)^{j} j!\frac{S (2k + i + j, j)}{(2k + i + j)!}, (5)

where S (p, q) denotes the Stirling numbers of the second kind.

From the explicit formula

S (p, q) = \frac{1}{q!} ∑_{l=0}^{q} (−1)^{l} \binom{q}{l} (q − l)^{p},

we immediately obtain our second main result.

Corollary 2. The Stirling coefficients have an exact representation of the form

a_{k} = \frac{(2k)!}{2^{k}k!} ∑_{i=0}^{2k} \binom{k + i − 1/2}{i}\binom{3k + 1/2}{2k − i}2^{i} ∑_{j=0}^{i} \binom{i}{j}\frac{(−1)^{j}}{(2k + i + j)!} ∑_{l=0}^{j} (−1)^{l} \binom{j}{l} (j − l)^{2k+i+j}.

(6)

To prove Theorem 1 we need some concepts. Let r ≥ 0 and a_{r} ≠ 0, let F (x) = ∑_{j≥r} a_{j}x^{j}/j! be a formal power series. The potential polynomials F_{n}^{(z)} in the variable z are defined by the exponential generating function

(\frac{a_{r}x^{r}/r!}{F (x)})^{z} = ∑_{n≥0} F_{n}^{(z)} \frac{x^{n}}{n!} .  (7)

For r ≥ 1, the exponential Bell polynomials B_{n,i} (0, …, 0, a_{r}, a_{r+1}, …) in an infinite number of variables a_{r}, a_{r+1}, … can be defined by

(F (x))^{i} = i! ∑_{n≥0} B_{n,i} (0, …, 0, a_{r}, a_{r+1}, …) \frac{x^{n}}{n!} .  (8)

The following theorem is due to Howard [2].

Theorem 3. If F_{n}^{(z)} is defined by (7) and B_{n,i} is defined by (8), then

F_{n}^{(z)} = ∑_{i=0}^{n} (−1)^{i} \binom{z + i − 1}{i}\binom{z + n}{n − i} (\frac{r!}{a_{r}} )^{i} \frac{n!i!}{(n + ri)!}B_{n+ri,i} (0, …, 0, a_{r}, a_{r+1}, …). (9)

Now we prove Theorem 1.

Proof of Theorem 1. Brassesco and Méndez showed that if

G (x) = 2\frac{e^{x} − x − 1}{x^{2}} = 2 ∑_{j≥0} \frac{x^{j}}{(j + 2)!},  (10)

then

a_{k} = \frac{1}{2^{k}k!}∂^{2k} (G^{− \frac{2k+1}{2}} ) (0) ,  (11)

where ∂^{k}f denotes the kth derivative of a function f. Define the polynomials G_{n}^{(z)} in the variable z by the following exponential generating function:

(\frac{1}{2} \frac{x^{2}}{e^{x} − x − 1})^{z} = ∑_{j≥0} G_{j}^{(z)} \frac{x^{j}}{j!} .  (12)

Inserting z = \frac{2k+1}{2} into this expression gives

∑_{j≥0} G(\frac{2k+1}{2})_{j} \frac{x^{j}}{j!} = (\frac{1}{2} \frac{x^{2}}{e^{x} − x − 1})^{\frac{2k+1}{2}} = (2\frac{e^{x} − x − 1}{x^{2}})^{− \frac{2k+1}{2}} = G^{− \frac{2k+1}{2}} (x) . (13)

On the other hand we have by series expansion

G^{− \frac{2k+1}{2}} (x) = ∑_{j≥0} ∂^{j} (G^{− \frac{2k+1}{2}} ) (0) \frac{x^{j}}{j!} .  (14)

Equating the coefficients in (13) and (14) gives

∂^{j} (G^{− \frac{2k+1}{2}} ) (0) = G(\frac{2k+1}{2})_{j} = G_{j}^{(k+ \frac{1}{2})}.

Now by comparing this with (11) yields

a_{k} = \frac{1}{2^{k}k!}G_{2k}^{(k+ \frac{1}{2})}.  (15)

Putting r = 2 an a_{r} = a_{r+1} = … = 1 into the formal power series F (x) = ∑_{j≥r} a_{j}x^{j}/j! gives F (x) = e^{x} − x − 1. And therefore the generated potential polynomials are

(\frac{x^{2}/2!}{e^{x} − x − 1})^{z} = (\frac{1}{2} \frac{x^{2}}{e^{x} − x − 1})^{z} = ∑_{j≥0} G_{j}^{(z)} \frac{x^{j}}{j!} .

According to Howard’s theorem we find

G_{n}^{(z)} = ∑_{i=0}^{n} (−1)^{i} \binom{z + i − 1}{i}\binom{z + n}{n − i}2^{i} \frac{n!i!}{(n + 2i)!}B_{n+2i,i} (0, 1, 1, …).  (16)

Now we derive an expression for the exponential Bell polynomials B_{n,i} (0, 1, 1, …) in terms of the Stirling numbers of the second kind:

i! ∑_{n≥0} B_{n,i} (0, 1, 1, …) \frac{x^{n}}{n!} = (F (x))^{i} = (e^{x} − x − 1)^{i}

= (−x + ∑_{l≥1} \frac{x^{l}}{l!} )^{i} = ∑_{j=0}^{i} \binom{i}{j} (−1)^{i−j} x^{i−j} (∑_{l≥1} \frac{x^{l}}{l!} )^{j}

= ∑_{j=0}^{i} \binom{i}{j} (−1)^{i−j} x^{i−j}j! ∑_{n≥0} S (n, j) \frac{x^{n}}{n!}

= ∑_{n≥0} ∑_{j=0}^{i} \binom{i}{j} (−1)^{i−j} j!S (n, j) \frac{x^{n+i−j}}{n!}

= i! ∑_{n≥0} \{\frac{n!}{i!} ∑_{j=0}^{i} \binom{i}{j} (−1)^{i−j} j!\frac{S (n − i + j, j)}{(n − i + j)!}\} \frac{x^{n}}{n!} .

Hence

B_{n,i} (0, 1, 1, …) = \frac{n!}{i!} ∑_{j=0}^{i} \binom{i}{j} (−1)^{i−j} j!\frac{S (n − i + j, j)}{(n − i + j)!}.  (17)

Thus we obtain

G_{n}^{(z)} = ∑_{i=0}^{n} \binom{z + i − 1}{i}\binom{z + n}{n − i}2^{i}n! ∑_{j=0}^{i} \binom{i}{j} (−1)^{j} j!\frac{S (n + i + j, j)}{(n + i + j)!}.  (18)

Substituting z = k + 1/2 and n = 2k into this expression yields

G_{2k}^{(k+ \frac{1}{2})} = ∑_{i=0}^{2k} \binom{k + i − 1/2}{i}\binom{3k + 1/2}{2k − i}2^{i} (2k)! ∑_{j=0}^{i} \binom{i}{j} (−1)^{j} j!\frac{S (2k + i + j, j)}{(2k + i + j)!}, (19)

hence by (15) we finally have

a_{k} = \frac{(2k)!}{2^{k}k!} ∑_{i=0}^{2k} \binom{k + i − 1/2}{i}\binom{3k + 1/2}{2k − i}2^{i} ∑_{j=0}^{i} \binom{i}{j} (−1)^{j} j!\frac{S (2k + i + j, j)}{(2k + i + j)!}.  (20)

This completes the proof of the theorem.

3 Acknowledgments

I am grateful to Lajos László, who drew my attention to the paper of Brassesco and Méndez. I also would like to thank the referees for their valuable remarks that help to improve the initial version of this paper.

References

[1] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Mcgraw–Hill Book Company, 1978, 218.

[2] F. T. Howard, A theorem relating potential and Bell polynomials, Discrete Math. 39 (1982), 129–143.

[3] G. Marsaglia and J. C. Marsaglia, A new derivation of Stirling’s approximation to n!, Amer. Math. Monthly 97 (1990), 826–829.

[4] J. M. Borwein and R. M. Corless, Emerging tools for experimental mathematics, Amer. Math. Monthly 106 (1999), 899–909.

[5] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel Publishing Company, 1974, p. 267.

[6] R. B. Paris and D. Kaminski, Asymptotics and Mellin–Barnes Integrals, Cambridge University Press, 2001, p. 32.

[7] S. Brassesco and M. A. Méndez, The asymptotic expansion for n! and Lagrange inversion

   formula, http://arxiv.org/abs/1002.3894.

2010 Mathematics Subject Classification: Primary 11B65; Secondary 11B73, 41A60. Keywords: asymptotic expansions, factorial, Stirling coefficients, Stirling’s formula, Stirling numbers.

(Concerned with sequences \underline{A001163}, \underline{A001164}, \underline{A008277}, \underline{A050211}, and \underline{A059022}.)

Received March 15 2010; revised version received June 17 2010. Published in Journal of Integer Sequences, June 21 2010.

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