On the stirling expansion into negative powers of a triangular number

Transcription

1 MATHEMATICAL COMMUNICATIONS 359 Math. Commun., Vol. 5, No. 2, pp ) On the stirling expansion into negative powers of a triangular number Cristinel Mortici, Department of Mathematics, Valahia University of Târgovişte, Bd. Unirii 8, Târgovişte , Romania Received April 30, 2009; accepted February 28, 200 Abstract. The aim of this paper is to answer an open problem posed by M. B. Villarino [arxiv: v2]. We also introduce a new accurate approximation formula for big factorials. AMS subject classifications: 40A25, 34E05, 33B5 Key words: Stirling s formula, Gosper s formula, asymptotic expansions. Introduction Maybe one of thost known and most used formula for approximation of big factorials is the Stirling s formula n! 2πn n+ 2 e n = σ n. It was first discovered by the French mathematician Abraham de Moivre ) with a missing constant), then the English mathematician James Stirling ) found the constant 2π. The Stirling s formula has important applications in many branches of science, being satisfactory in probability theory, statistical physiscs, or mechanics, while in purathematics more accurate approximations are required. In fact, the Stirling s formula is the first approximation of the following series n! 2πn n+ 2 e n + 2n + 288n 2 39 ) 5840n n For details see [, p. 257]. The gamma function Γ is defined for x > 0, by Γ x) = 0 t x e t dt and it is a natural extension of the factorial function, since Γ n + ) = n!, for all n =, 2, 3,.... The factorial and gamma function are related with the harmonic sum Corresponding author. address: cmortici@valahia.ro C. Mortici) c 200 Department of Mathematics, University of Osijek

2 360 C. Mortici H n = n in the sense we describe next. The digamma function ψ defined as the logarithmic derivative of the gamma function satisfies the recurrence relation ψ x) = d dx ln Γ x)) = Γ x) Γ x), ψ x + ) = ψ x) + x ; thus, implying the formula ψ n) = H n γ, where γ = is the Euler-Mascheroni constant. In 755, the Swiss mathematician Leonhard Euler ) found the asymptotic expansion for H n, B k H n ln n + γ n k, where B k denotes the k th Bernoulli number [, p. 804]. Ramanujan [3, p. 52] found the asymptotic expansion of H n into powers of the reciprocal of the n th triangular number m = nn+) 2. Precisely, as n approaches infinity, H n 2 ln 2m) + γ + 2m 20m m 3 680m m M. B. Villarino finishes his work [22] with the remark that it would be interesting to develop an expansion for n! into powers of m, that is, a new Stirling expansion of the form n! n ) n 2πn e + k= k= a k m k ). ) Motivated by this fact, in this paper we try to introduce a new method for constructing such a series. Until now, we were able to give the first term of that expansion. More precisely, we propose the approximation ) ) n π 2 2πn e 2 6 n k= k 2 ) = µ n, 2) e 48m which gives better results than Stirling s formula and other known results. As it follows from our study, a performant series ) can be constructed only if an additional factor of the form is considered. exp 2 π 2 n 6 k= )) k 2

3 2. The results On the Stirling expansion into negative powers 36 In what follows, we need the following result, which is a measure of the rate of convergence. Lemma. If λ n ) n is convergent to zero and there exists the limit with k >, then there exists the limit: lim n nk λ n λ n+ ) = l R, 3) lim n nk λ n = l k. 4) Limit 4) offers the rate of convergence of the sequence λ n ) n and we can see that the sequence λ n ) n converges faster to zero, as the value k satisfying 3) is greater. This Lemma was first used by Mortici [6 20] for constructing asymptotic expansions, or to accelerate some convergences. For proof and other details, see, e.g., [7], or [8]. First, let us define the sequence λ n ) n by the Stirling s approximation n! = n ) n 2πn exp λn. 5) e We have λ n λ n+ = n + ) ln + ) 2 n and from routine calculations we obtain lim n n 2 λ n λ n+ ) = 2. By applying Lemma, the sequence λ n ) n converges to zero as n. Now, if we are interested in constructing an approximation of the form ), then we expect that already the first approximation ) n 2πn + a ) 6) is much better than the Stirling s formula. In our language introduced here, we ask that the sequence ω n ) n defined by n! = n ) n 2πn + a ) exp ω n 7) should be faster convergent to zero than the sequence λ n ) n, that is, it should converge to zero faster than n. From 7), we deduce that ω n ω n+ = or, using computer software, ω n ω n+ = 2n 2 n + ) ln + ) + ln + 2a n+)n+2) 2 n + 2a nn+) 4a + ) 2 n 3 + 2a + 3 ) ) 40 n 4 + O n 5. 8)

4 362 C. Mortici We have lim n n 2 ω n ω n+ ) = 2. From Lemma, lim n nω n = 2, independently of the value a, and consequently, approximation 6) cannot bade better than the Stirling s formula. In other words, a series of the form ) cannot be constructed in the background we present later. We can improve the rate of convergence by annihilating the term 2n. After a 2 careful analysis, we introduce a class of approximations of the form ) ) n π 2 2πn e 2 6 n k= k 2 + a ), 9) with m = nn+) 2 and a R. Let us define the sequence τ n ) n by n! = n ) n π 2 ) 2πn e 2 6 n k= k 2 + a ) exp τ n, 0) associated with the approximation formula 9). Now we are in the position to state the following Theorem. Let n! = n ) [ ] [ n 2a π 2 2πn + exp e n n + ) 2 n 6 i= ) ] i 2 + τ n for n N. Then for a 48 and for a = 48. lim n 2 ) τ n = 2a + ) 0 n 24 lim n 3 ) 7 τ n = n 20 Proof. From 0), we have τ n τ n+ = or, again using a computer software, τ n τ n+ = 4a + 2 n + ) ln + ) + ln + 2a n+)n+2) 2 n + 2a nn+) ) n 3 + 2n 2, 2a + 3 ) ) 40 n 4 + O n 5. Now i) follows immediately from Lemma. In case a = 48, we obtain τ n τ n+ = 7 ) 40n 4 + O n 5, which justifies the statement ii).

5 On the Stirling expansion into negative powers 363 This Theorem shows that the best approximation of the form 9) is 2), which is obtained for a = 48. The corresponding asymptotic formula ) [ ] [ )] n 2a π 2 n 2πn + exp e n n + ) 2 6 i 2, n is better than some known results. Some works about the approximating n!, gamma and related functions investigate this problem from the viewpoint of inequalities and logarithmically completonotonicity. See, for example [6 2]. 3. Conclusions Some of slightly more accurate approximation formula than Stirling s result, are the following: n! ) n+/2 n + /2 2π W. Burnside [4]). ) e ) n+/2 2π n + n! C. Mortici [6]). 2) e e Better results were recently established by N. Batir [2] 2πn n+ e n n! 3) n /6 i= and R. W. Gosper [5] n! 2π n + ) n ) n = γn. 4) 6 e The best approximations of )-4) is the Gosper s formula 4). The following numerical computations show the great superiority of our new formula 2) over the Gosper s formula 4). For the sake of completeness, we also consider the Stirling s approximation n! σ n. n! n! σ n n! γ n µ n n! Acknowledgement The author is grateful to the anonymous referees whose comments and corrections led to a significant improvement of the initial manuscript.

6 364 C. Mortici References [] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 964. [2] N. Batir, Sharp inequalities for factorial n, Proyecciones, ), [3] B. Berndt, Ramanujan s Notebooks, Springer, New York, 998. [4] W. Burnside, A rapidly convergent series for log N!, Messenger Math. 4697), [5] R. W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci ), [6] C. Mortici, An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math ), [7] C. Mortici, Product approximations via asymptotic integration, Amer. Math. Monthly 7200), [8] C. Mortici, New approximations of the gamma function in terms of the digamma function, Appl. Math. Lett ), [9] C. Mortici, New sharp bounds for gamma and digamma functions, An. Ştiinţ. Univ. A. I. Cuza Iaşi Ser. N. Matem., to appear. [0] C. Mortici, Completely monotonic functions associated with gamma function and applications, Carpathian J. Math ), [] C. Mortici, The proof of Muqattash-Yahdi conjecture, Math. Comput. Modelling 5200), [2] C. Mortici, Monotonicity properties of the volume of the unit ball in R n, Optimization Lett. 4200), [3] C. Mortici, Sharp inequalities related to Gosper s formula, C. R. Math. Acad. Sci. Paris ), [4] C. Mortici, A class of integral approximations for the factorial function, Comput. Math. Appl ), [5] C. Mortici, Best estimates of the generalized Stirling formula, Appl. Math. Comput ), [6] C. Mortici, Very accurate estimates of the polygamma functions, Asymptot. Anal ), [7] C. Mortici, Improved convergence towards generalized Euler-Mascheroni constant, Appl. Math. Comput ), [8] C. Mortici, A coincidence degree for bifurcation problems, Nonlin. Anal ), [9] C. Mortici, A quicker convergence toward the γ constant with the logarithm term involving the constant e, Carpathian J. Math ), [20] C. Mortici, Optimizing the rate of convergence in some new classes of sequences convergent to Euler s constant, Anal. Appl. Singap.) 8200), [2] F. Qi, Bounds for the ratio of two gamma functions, available at [22] M. B. Villarino, Ramanujan s Harmonic Number Expansion into Negative Powers of a Triangular Number, J. Inequal. Pure and Appl. Math ), Article 89.