The quotient of gamma functions by the psi function

The quotient of gamma functions by the psi function

Cristinel Mortici

Valahia University of Târgoviste, Department of Mathematics, Bd. Unirii 18, 130082 Târgoviste, Romania. E-mails: [e-mail]; [e-mail]

ABSTRACT

The aim of this paper is to construct the asymptotic series of the ratio of gamma functions by Kershaw then to deduce some sharp estimates.

Mathematical subject classification: 33B15, 05A16, 26D15.

Key words: gamma function, digamma function, inequalities, asymptotic series, rate of convergence.

1 Introduction

The gamma function, defined for every x > 0 by the formula

is of great importance in pure and applied mathematics and in consequence it attracted the attention of many authors. Some preoccupations are now directed to the problem of estimating the ratio

in terms of the digamma function ψ = Γ^'/ Γ. The starting point of this problem can be considered the work of Gautschi [3], who proved

for every 0 < β < 1 and x = 1,2,3,.... This inequality (1.1) has found great interest, and several papers were subsequently published, for instance, by Erber [2], Kečkić and Vasić [4], Laforgia [6], Watson [26], or Zimering [27]. Furthermore, Lorch [9] found interesting connections of this inequality with the theory of ultraspherical polynomials.

Kershaw [5] improved (1.1), showing

for every 0 < β < 1 and x > 1 – β. This inequality (1.2) is now known as the second Kershaw inequality.

Merkle [10] exploited some convexity arguments to improve the left-handside of (1.2). By [10, Corollary 3], for every 0 < β < 1 and x > 0,

Obviously, the main purpose of inequalities of type (1.1)-(1.3) is to obtain increasingly strong estimates of the ratio Q(x, β) , as x approaches infinity. We consider that the natural way to attack such problems is to provide estimates of Q(x, β) using asymptotic analysis theory. Although inequalities following by different convexity arguments are of intinsec beauty, they do not offer estimates to any expected accuracy.

In the first part of this paper we prove that the best approximation of the form

where m is a real parameter, is obtained indeed for m = β/2.

In asymptotic theory, whenever an approximation of type (1.4) is given, there is a tendency to improve it by using an entire series of the form

In general, such a series does not converge, but truncated of only a few terms, it provides approximations of any desired accuracy.

One of the main result of this paper is the construction of the asymptotic expansion (1.5). We give a systematically way to find the coefficients a_j, while the first coefficients are the following

Numerical computations show that Q(x, β) becomes closer to the righ-hand side of (1.2), for large values of x. In consequence, strong inequalities of the form

are obtained only if the quantities m₁(x, β) and m₂(x, β) becomes closer to 1, as x approaches infinity.

Motivated by these remarks, we use the first terms of expansion (1.5) todeduce the inequality of type (1.7) with

for every 0 < β < 1 and sufficiently large integer x.

2 The best approximation Q(x, β ) ≈ exp ( β ψ (x + m))

We show in this section that the best approximation of the form (1.4) is obtained for m = β/2. One method to estimate the accuracy of an approximation of type (1.4) is to introduce the relative error sequence w_n by the relations

and to consider an approximation (1.4) better when w_n faster converges to zero.

A powerful way to measure the rate of convergence is the following

Lemma 1. If ( ω_n)_{n> 1} is convergent to zero and

with k > 1, then

This lemma was first used by Mortici [11]-[25] to construct asymptotic expansions and to improve some convergences. For complete proof of Lemma 1,see [11].

By (2.1), we have

w_n = ln Q(n, β) – β ψ (n + m)

and as we are interested to compute a limit of the form (2.2), we write the difference

as a power series into n^-1. Using a computer software, we get

Now we can apply Lemma 1 to establish the following

Theorem 1. (i) If m ≠ β/2, then w_n converges as n^-1, since

(ii) If m = β/2, then w_n converges as n^-2, since

Finally, the best approximation (1.4) appears when the maximum rate of convergence n^-2 of w_n is attained, namely in case m = β/2.

As a direct consequence of this fact, we mention that every other inequalityof type Q(x, β) < ( > ) exp( β ψ (x + m) ), with m ≠ β/2 (largely studied inthe literature) becomes weak for large values of x.

3 The asymptotic expansion of Q(x, β )

In this section, the main tool for constructing the asymptotic series (1.5) is the method introduced by Mortici in the recent paper [11].

Let us consider the approximation f(n) ≈ g( n), in the sense that the ratio of f(n) and g(n) tends to 1 as n approaches infinity and assume that

Then the following asymptotic series holds

where the coefficients a_j are given by the infinite triangular system

For proofs and other details, see [11].

Lemma 2. For

we have

Proof. We have

Now, with

the first equations of the infinite system (3.1) becomes

with the solution (1.6).

4 Sharp bounds of Q(x, β )

As usually, we try to compare Q(x, β) with approximations obtained by truncation the corresponding asymptotic expansion (1.5). More precisely, we prove that for large values of x, we have the following double inequality that improves much the Gautschi-Kershaw inequalities (1.1)-(1.3)

The left-hand side of (4.1) is valid for β > , while it reverses for β <.

Theorem 2. There exists n₀ such that for every integer x > n₀ and 0 < β < 1, we have

Proof. The sequence

converges to zero. We prove that for some n₀, the sequence (a_n)_n>n0 is strictly decreasing and consequently, a_n > 0, for every n > n₀ and the theorem is proved.

In this sense, we have a_n+1 – a_n = f(n) , where

We have

where Q(x) = 24(1 – β)x⁴ + ... – β³ is a fourth degree polynomial in x.

As the leading coefficient of Q is positive, we have Q > 0, on an interval of the form [n₀, ∞) and consequently, f is strictly increasing on [n₀, ∞). But f( ∞) = 0, so f(x) < 0, for every x > n₀. Finally, the sequence (a_n) _n>n0 is strictly decreasing, thus a_n > 0 and the proof is completed.

Theorem 3. a) There exists n₁such that for every integer x > n₁and β ∈, we have

b) There exists n₂ such that for every integer x > n₂ and β ∈,we have

Proof. The sequence

converges to zero. We prove that starting with some index, the sequence b_n is strictly decreasing (respective increasing) and consequently, b_n > 0, (respective b_n < 0), and the theorem is proved.

In this sense, we have b_n+1 – b_n = g(n) , where

We have

where P is a fifth degree polynomial in x,

P(x) = (120 β – 66 β² – 40)x⁵ + ... + (3 β³ – 3 β⁴) .

Now, the leading coefficient of P is negative if β ∈ , respective positive, if β ∈ . In consequence, if β ∈ , then P < 0 on an interval of the form [n₁, ∞), and if β ∈ , then P > 0 on an interval of the form [n₂, ∞).

As g is proved to be strictly increasing on [n₁, ∞) (respective strictly decreasing on [n₂, ∞)) and g( ∞) = 0, it results that g < 0 on [n₁, ∞) (respective g > 0 on [n₂, ∞)). Finally, if β ∈ , then (b_n)_n>n1 is strictly decreasing, while if β ∈ , then (b_n)_n>n2 is strictly increasingand the theorem is proved.

5 Q(x, 1/2) and the Wallis ratio

The general results obtained in the previous sections can be extended, forparticular cases of β ∈ (0,1), if we think that the polynomials involved in (4.2) and (4.5) becomes simpler.

More precisely, the privileged value β = 1/2 is of great interest, since Q(x, 1/2) is close related with the Wallis ratio and with the Wallis sequence

Moreover, the Wallis ratio appears in the problem of estimating the volume and the surface area of the unit ball in

ⁿ. See, e.g., [1, p. 258, Relation 6.1.49], or [17].

Gautschi [3] proved

then Kazarinoff [8, pp. 47-48 and pp. 65-67] showed

We obtain here sharp estimates for Q(x,1/2) , using the asymptotic expansion (1.5), which becomes

The double inequality (4.1) becomes

but we able now to prove the following stronger result.

Theorem 4. For every positive integer x, we have

Proof. We prove that the convergent to zero sequences

and

are strictly increasing, thus u_n < 0 and v_n < 0.

Let u_n+1 – u_n = s(n), v_n+1 – v_n = t(n), where

and

The functions s(x) and t(x) are strictly decreasing on [1, ∞), since

and

Moreover, s( ∞) = t( ∞) = 0, so s > 0 and t > 0 on [1, ∞). Finally, u_n and v_n are strictly increasing and the theorem is proved.

Acknowledgments. This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, project number PN-II-ID-PCE-2011-3-0087.

Received: 23/VIII/10.

Accepted: 11/III/11.

#CAM-250/10.

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Publication Dates

• Publication in this collection
  06 Jan 2012
• Date of issue
  2011

History

• Received
  23 Aug 2010
• Accepted
  11 Mar 2011