On-line version ISSN 1807-0302
Comput. Appl. Math. vol.31 no.1 São Carlos 2012
Chao-Ping ChenI; Cristinel MorticiII, *
ISchool of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City 454003, Henan Province, People's Republic of China E-mail: firstname.lastname@example.org
IIValahia University of Târgoviste, Department of Mathematics, Bd. Unirii 18, 130082 Târgoviste, Romania E-mail: email@example.com
Let ψ denote the psi (or digamma) function. We determine the values of theparameters p, q and r such that
ψ(n) ≈ ln(n + p) -
is the best approximations. Also, we present closer bounds for psi function, which sharpens some known results due to Muqattash and Yahdi, Qi and Guo, and Mortici.
Mathematical subject classification: 33B15, 26D15.
Key words: psi function, polygamma functions, rate of convergence, approximations.
The gamma function is usually defined for x > 0 by
The logarithmic derivative of the gamma function:
is known as the psi (or digamma) function. The successive derivatives of the psi function ψ(x):
are called the polygamma functions.
The following asymptotic formula is well known for the psi function:
(see [1, p. 259]), where
are the Bernoulli numbers.
Recently, the approximations of the following form:
were studied by Muqattash and Yahdi . They computed the error
and then the approximation (2) was compared with the approximation obtained by considering the first two terms of the series (1), that is
Very recently, the family (2) was also discussed by Qi and Guo . One of their main results is the following inequality on x ∈ (0, ∞):
where γ = 0.577215... is the Euler-Mascheroni constant.
In the final part of the paper , the authors wonder whether there are profitable constants a ∈ [0, 1] and b ∈ [1, 2] for which better approximations of the form
can be obtained. Mortici  solved this open problem and proved that the best approximations (4) appear for
Moreover, the author derived from [3, Theorem 2.1] the following symmetric double inequality: For x > = 0.40824829...,
This double inequality is more accurate than the estimations (3) of Qi and Guo.
We define the sequence by
We are interested in finding the values of the parameters p, q and r such that is the fastest sequence which would converge to zero. This provides the best approximations of the form:
Our study is based on the following Lemma 1, which provides a method for measuring the speed of convergence.
Lemma 1 (see  and ). If the sequence converges to zero and if there exists the following limit:
Theorem 1. Let the sequence be defined by (6). Then for
The speed of convergence of the sequence is given by the order estimate O(n-4) as n → ∞.
Proof. First of all, we write the difference vn - vn + 1 as the following power series in n-1:
According to Lemma 1, the three parameters p, q and r, which produce the fastest convergence of the sequence are given by (10)
that is, by (8) and (9). We thus find that
Finally, by using Lemma 1, we obtain the assertion (1) of Theorem 1.
Solutions (8) and (9) provide the best approximations of type (7):
Theorem 2 below presents closer bounds for psi function.
Theorem 2. For x > = 1.23394491..., then
Proof. The lower bound of (13) is obtained by considering the function F defined by
We conclude from the asymptotic formula (1) that
It follows form [2, Theorem 9] that
Differentiating F(x) with respect to x and applying the second inequality in (14) yields, for x > ,
Therefore, F'(x) < 0 for x > . This leads to
This means that the first inequality in holds for x > .
The upper bound of (13) is obtained by considering the function G defined by
We conclude from the asymptotic formula (1) that
Differentiating G(x) with respect to x and applying the first inequality in (14) yields, for x > 0,
Therefore, Q(x) > 0 and G'(x) > 0 for x > x2. This leads to
This means that the second inequality in (13) holds for x > 0.158650823....
Some computer experiments indicate that for x > 2.30488055, the lower bound in (13) is sharper than one in (5). For x > 0.5690291018, the upper bound in (13) is sharper than one in (5).
The inequality (13) provides the best approximations:
Acknowledgements. The work of the second author was supported by agrant of the Romanian National Authority for Scientific Research, CNCS -UEFISCDI, project number PN-II-ID-PCE-2011-3-0087.
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* Corresponding author.