On values of the psi function

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@article{Adam_2017,
   doi = {10.17512/jamcm.2017.1.01},
   url = {https://doi.org/10.17512/jamcm.2017.1.01},
   year = 2017,
   publisher = {The Publishing Office of Czestochowa University of Technology},
   volume = {16},
   number = {1},
   pages = {7--18},
   author = {Marcin Adam and  Bożena Piątek and  Mariusz Pleszczyński and  Barbara Smoleń and  Roman Wituła},
   title = {On values of the psi function},
   journal = {Journal of Applied Mathematics and Computational Mechanics}
}

TY  - JOUR
DO  - 10.17512/jamcm.2017.1.01
UR  - https://doi.org/10.17512/jamcm.2017.1.01
TI  - On values of the psi function
T2  - Journal of Applied Mathematics and Computational Mechanics
JA  - J Appl Math Comput Mech
AU  - Adam, Marcin
AU  - Piątek, Bożena
AU  - Pleszczyński, Mariusz
AU  - Smoleń, Barbara
AU  - Wituła, Roman
PY  - 2017
PB  - The Publishing Office of Czestochowa University of Technology
SP  - 7
EP  - 18
IS  - 1
VL  - 16
SN  - 2299-9965
SN  - 2353-0588
ER  - 

Adam, M., Piątek, B., Pleszczyński, M., Smoleń, B., & Wituła, R. (2017). On values of the psi function. Journal of Applied Mathematics and Computational Mechanics, 16(1), 7-18. doi:10.17512/jamcm.2017.1.01

Adam, M., Piątek, B., Pleszczyński, M., Smoleń, B. & Wituła, R., 2017. On values of the psi function. Journal of Applied Mathematics and Computational Mechanics, 16(1), pp.7-18. Available at: https://doi.org/10.17512/jamcm.2017.1.01

[1]M. Adam, B. Piątek, M. Pleszczyński, B. Smoleń and R. Wituła, "On values of the psi function," Journal of Applied Mathematics and Computational Mechanics, vol. 16, no. 1, pp. 7-18, 2017.

Adam, Marcin, Bożena Piątek, Mariusz Pleszczyński, Barbara Smoleń, and Roman Wituła. "On values of the psi function." Journal of Applied Mathematics and Computational Mechanics 16.1 (2017): 7-18. CrossRef. Web.

1. Adam M, Piątek B, Pleszczyński M, Smoleń B, Wituła R. On values of the psi function. Journal of Applied Mathematics and Computational Mechanics. The Publishing Office of Czestochowa University of Technology; 2017;16(1):7-18. Available from: https://doi.org/10.17512/jamcm.2017.1.01

Adam, Marcin, Bożena Piątek, Mariusz Pleszczyński, Barbara Smoleń, and Roman Wituła. "On values of the psi function." Journal of Applied Mathematics and Computational Mechanics 16, no. 1 (2017): 7-18. doi:10.17512/jamcm.2017.1.01

ON VALUES OF THE PSI FUNCTION

Marcin Adam, Bożena Piątek, Mariusz Pleszczyński, Barbara Smoleń, Roman Wituła

Institute of Mathematics, Silesian University of Technology
Gliwice, Poland
marcin.adam@polsl.pl, bozena.piatek@polsl.pl, mariusz.pleszczynski@polsl.pl roman.witula@polsl.pl

Received: 20 December 2016; accepted: 16 March 2017

Abstract. In the present paper, values of the psi function for many arguments connected with the golden ratio and Fibonacci numbers are determined or given in alternative form. Moreover, some integral representation of the psi function is found. This is a potential calculation base of values of the psi function for powers of argument. We also note that this integral representation gives better numerical estimation of values of the psi function than the respective Legendre’s integral formula.

MSC 2010: 33E50

Keywords: psi function, golden ratio, Fibonacci numbers

1. Introduction

The present paper concerns the values of the psi function in arguments connected with the golden ratio (and in consequence with Fibonacci and Lucas numbers) as well as with powers of argument in general form. The main reason for our interest in this area is a lack of results of this type in the literature [1-9]). Of course, we would like to fill this gap. Fundamental results obtained here (Theorems 2 and 3) substantially enrich (and complete) the state of knowledge of the considered topic. Moreover, these results also show that there are still some new areas of applications of the golden ratio, Fibonacci numbers and corresponding recurrence equations (for example as an another and simultaneously new area of such application - see the contents of our recent published paper [10]). We emphasize that experience acquired while writing the book [7] - particularly by the last author, were important in creating this paper. They also had a direct impact on the content of research taken here.

2. Technical introduction

The technical realization of our project requires a number of preparations and it forced us to derive many additional relations. Let us start with recalling basic notions and facts related to the psi function.

The psi function is defined to be the logarithmic derivative of the gamma function:

It constitutes one of the most important special functions, which occur today innearly all fundamental analytic areas of mathematics. Let us recall some basic identities for the psi function which will be applied in the sequel. The majority of them can be found in [7] (see also [1, 2]). Namely, we have

where is the Euler constant (see [7]). Moreover,will denote here the golden ratio (so ).

The following lemma is crucial for our considerations, especially in the proof of Theorem 3. Let us emphasize that this technical result has a wide range of applications (see additionally [11] or [12]), and as far as identities a) - d) are known, the result from point e) seems to be original. Although identities a) - d) are not new, they are very often presented in different forms. So, for the reader’s convenience, we have decided to add the proof of the point c) - the most intriguing case among the presented ones.

Lemma 1. The following identities hold true:

a)

b)

c)

d)

e) Let p and suppose that

for sufficiently large rwhere

,

i.e. is the sequence of convolution type sums connected with polynomial p and a given sequence .

Proof. a) - d) All proofs can be done by using the Binet's formulae for and , respectively. We only prove identity c). First, observe that

where . We note that so

for and , which implies

which for implies c) because . The formula c) does not hold for.

,

from which the relation (4) follows directly.

3. Main result for powers of argument

The main goal of this section is to present the integral type formula for the psi function for powers of argument. As it is shown below, our integral formula (5) is better than the classic Legendre’s one (see (6)).

Theorem 2. The function psi satisfies the following integral relation:

where and

Proof. We note that

by applying the L’Hospital’s rule twice. Let us use the following Ramanujan integral (see [13, 14]):

Now, put and . Summing up these equalities side by side for , we get

which is the desirable equality (5).

Referring to equation (5) one might ask whether it would be better to use the following Legendre's formula (see [7], section 4.3)

to calculate ? It is easy to check that if we replace by and is sufficiently large, then the numerical convergence of the integral from (6) is worse than convergence of the integral from (5). The following relations explain it better:

and

Therefore it is better to apply the formula (5) to find

We propose comparing the integral relation (5) with the asymptotic formula from the point f) of Theorem 3.

It is very surprising that for the psi function there is another couple of formulae of type (5) and (6), namely

The proof of the formula (8) can be found in [15]. Formulae (7) and (8) differ from (5) and (6) by the speed of convergence of the respective functional series occurring there. The identity (7) and especially the identity (8) by appropriate iterations of these formulae can be used to calculate the value of for large values of .

4. The values of the psi function in arguments connected with the golden ratio and Fibonacci numbers

The values of the psi function for many different arguments connected with the golden ratio (especially for powers of ) as well as Fibonacci and Lucas numbers will be shown in our second main theorem.

Theorem 3. The following identities hold:

a)

b)

c)

where denotes the -th harmonic number

d)

Moreover, if then

which yields

On the other hand if 2∤ n then

which leads to

e) we have

and

which implies that

f) and

for each In the sequel we obtain

g)

where

– if then and

– if 2 ∤n then and

Proof.

a)

b)

since

c)

e) the relations follow from the known identity (see also [16, 17]):

and identities (1) and (3).

f) There are constants (for instance ) and (for instance ) (see [18] or [19]) such that

for Then for we get

i.e.

Moreover, we have

when ever which finishes the proof.

Corollary 4. We have

where since

where , i.e.

where , i.e. .

5. Final remark

In the previous theorem, arguments of the psi function were reduced to numbers contained in the interval . This operation is especially interesting from the algebraic point of view, but it is not proper from the numerical point of view (asimilar situation takes place for the gamma function - it is better to replace a given argument by the respective large value of the argument, with respect to the Stirling formula, which plays not only an asymptotic formula role but even the exact role in approximation of values of the gamma function). The values of the psi function could be calculated using the following three inequalities of the asymptotic type which hold true for all (and which especially play a crucial role for large values of ):

(see [20, Theorem C]),

(see [18]),

(see [21]), or using the asymptotic series of function as :

(see [5] and [6]).

6. Conclusions

In the paper, several formulae for values of the psi function are presented. Some of them are of the asymptotic type, and some of them are of the symbolic-arithmetic type. It seems that the planned goal has been accomplished, and even more (as in the case of the integral formula for the psi function for powers of argument which, from a numerical point of view, is better than the respective Legendre’s integral formula). It is important that in comparison with many formulae for the psi function (also recently proven), relations obtained in the present paper complement the existing directory of these formulae well and substantially.

References

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