Direct spreading measures of Laguerre polynomials
The direct spreading measures of the Laguerre polynomials , which quantify the distribution of its Rakhmanov probability density along the positive real line in various complementary and qualitatively different ways, are investigated. These measures include the familiar root-mean-square or standard deviation and the information-theoretic lengths of Fisher, Renyi and Shannon types. The Fisher length is explicitly given. The Renyi length of order (such that ) is also found in terms of by means of two error-free computing approaches; one makes use of the Lauricella function , which is based on the Srivastava-Niukkanen linearization relation of Laguerre polynomials, and another one which utilizes the multivariate Bell polynomials of Combinatorics. The Shannon length cannot be exactly calculated because of its logarithmic-functional form, but its asymptotics is provided and sharp bounds are obtained by use of an information-theoretic optimization procedure. Finally, all these spreading measures are mutually compared and computationally analyzed; in particular, it is found that the apparent quasi-linear relation between the Shannon length and the standard deviation becomes rigorously linear only asymptotically (i.e. for ).
keywords:Orthogonal polynomials, Laguerre polynomials, spreading lengths, computation of information measures, Shannon entropy, Renyi entropy, Fisher information, Bell polynomials, Lauricella functions.
Msc:33C45, 94A17, 62B10, 65C60
The Laguerre polynomials are real hypergeometric polynomials orthonormal with respect to the weight function on the interval . Their algebraic properties (orthogonality relation, three-term recurrence relation, second-order differential equation, ladder relation,…) are simple and widely known chihara_78 (); nikiforov_88 (); andrews_99 (), what has enabled to describe a great deal of scientific and technological phenomena. The Laguerre polynomials are known to play a crucial role not only in applied and computational mathematics chihara_78 (); andrews_99 (); ismail_05 (); temme_96 (); nikiforov_88 (); chen:mcm03 (); srivastava:ass88 (); hounkonnon:ame00 (); srivastava_85 (), mathematical physics nikiforov_88 (); crandall:jcp85 (), combinatorics foata_82 (); desainte:lnm85 (); foata:sjdm88 (); askey_jct78 (), information theory yanez:jmp08 (); dehesa:jcam01 (); sanchezmoreno:preprint09 (); aptekarev:jcam09 (); dehesa:jcam06 (); sanchezruiz:jcam05 (); aydmer_arxiv06 (), quantum algebras srivastava:ass88 (); dattoli_arxiv07 (), asymptotics aptekarev:jcam09 (); borwein:sjna08 (); aptekarev:rassm95 (); dehesa:jmp98 (); sanchezruiz:jcam00 () and theory of special functions temme_96 (); nikiforov_88 (); yanez:jmp08 (); dehesa:jcam01 (); sanchezmoreno:preprint09 (); aptekarev:jcam09 (); srivastava:mcm03 (); dehesa:jcam06 (); sanchezruiz:jcam05 (); niukkanen:jpa85 (); devicente_04 (), but also in non-relativisitic, relativistic and supersymmetric quantum mechanics galindo_pascual_90 (); bagrov_90 (); cooper_01 (); flugge_71 (); niukkanen:jpa85 (), atomic and molecular physics sanchezruiz:jcam03 (); bransden_03 (); dehesa:mp06 (); aydmer_arxiv06 () and D-dimensional physics wang:fpl02 (); nieto_ajp79 (); dong:mpl05 (). Let us just mention that the Laguerre polynomials control to a great extent the wavefunctions which describe the quantum states of one and many-body systems with a great diversity of quantum-mechanical potentials galindo_pascual_90 (); sanchezmoreno:preprint09 (); omiste:jmc09 (); flugge_71 (); sanchezruiz:jcam03 (); dunkl:aa03 (); dehesa:mp06 (); nieto_ajp79 (); sanchezruiz:jcam00 (); aydmer_arxiv06 (); gangopadhyay_jpa98 (). The Coulomb and Morse potentials are only two particularly relevant cases in atomic and molecular physics (see e.g. bagrov_90 (); flugge_71 (); crandall:jcp85 (); bransden_03 (); dunkl:aa03 ()) as well as in -dimensional physics wang:fpl02 (); alves:jpa88 (), where the radial wavefunctions are controlled by Laguerre polynomials.
In this work we study the spreading measures of Laguerre polynomials , which quantify the distribution of its associated probability density
where is a constant such that is normalized to the unity. Heretoforth this distribution is called the Rakhmanov density of the polynomials, in honour to the pioneering work rakhmanov:mus77 () of this mathematician who has shown that this density governs the asymptotic behaviour of the ratio of two orthogonal polynomials with contiguous orders and . Physically, this probability density characterizes the stationary states of a large class of quantum-mechanical potentials sanchezmoreno:preprint09 (); bagrov_90 (); yanez:jmp08 (); cooper_01 (); nikiforov_88 (); omiste:jmc09 (). The most familiar spreading measure is the simple root-mean-square or standard deviation
where the expectation value of a function is defined by
The information-theoretic-based spreading measures of the Laguerre polynomials are not so well known. We refer to the Fisher information
to the Renyi entropy of order
and its limit, the Shannon entropy
which measure the distribution of the Laguerre polynomial all over the orthogonality interval without reference to any specific point of the interval, so providing alternative and complementary measures for the spreading of the Laguerre polynomials. The knowledge of these measures and some quantum-mechanical applications is reviewed in Ref. dehesa:jcam01 () up to 2001. Their behaviour for large and fixed has been recently surveyed aptekarev:jcam09 ().
Since the Fisher, Renyi and Shannon measures of a given density have particular units, which are different from that of the variable , it is much more useful to use the related information-theoretic lengths hall:pra99 (); hall:pra00 (); hall:pra01 (); sanchezmoreno:jcam09 (); guerrero:preprint10 (); namely, the Fisher length given by
and the -order Renyi and Shannon lengths defined by
respectively. Following Hall hall:pra99 (); hall:pra00 (); hall:pra01 (), these three quantities together with the standard deviation will be referred as the direct spreading measures of the density because they share the following properties: linear scaling under adequate boundary conditions, same units as the variable, and vanishing when the density tends to a delta density. Moreover, they have an associated uncertainty property hall:pra99 (); hall:pra00 (); hall:pra01 (); dehesa:mp06 () and fulfil the inequalities
Here we will investigate the direct spreading measures of the Laguerre polynomials mentioned above. First, in Section II, we give the known values of the ordinary moments, the standard deviation and the Fisher length of these polynomials. Second, in Section III, the entropic moments and the Renyi lengths are computed by use of two different approaches; one makes use of the Srivastava-Niukkanen linearization relation srivastava:mcm03 () of Laguerre polynomials, and another one based on the combinatorial multivariate Bell polynomials riordan_80 (); sanchezmoreno:jcam09 (). Third, in Section IV, the asymptotics of the Shannon length is given and some sharp bounds to this measure are found. Then, in Section IV, all the four spreading measures are computationally discussed. Finally, some open problems and conclusions are given.
2 Ordinary moments, standard deviation and Fisher length
In this section the known values for the moments-around-the-origin , the standard deviation and the Fisher length of the Laguerre polynomials are given. Let us start writing the orthonormality relation
for the orthonormal Laguerre polynomials
Then, the moment-around-the-origin of order is defined, according to Eq. (3), by
one finds that
so that the Fisher length of these polynomials has, according to (6), the value
It is worth remarking that the inequality is clearly satisfied.
3 Renyi lengths
In this section the Renyi lengths of the Laguerre polynomials defined by Eq. (7) will be computed by two different approaches: an algebraic approach which is based on the Srivastava-Niukkanen linearization relation srivastava:mcm03 (), and a combinatorial method which utilizes the multivariate Bell polynomials riordan_80 (). Let us advance that the Renyi integrals are computed only for the half-integer values of . It is worth noting that the final formulas cannot clearly be extended to all real values of , which explains why we cannot take limits to get the Shannon entropy in a straightforward manner.
are the frequency or entropic moments of the Rakhmanov density (1) of the Laguerre polynomials. In spite of the efforts of numerous researchers andrews_99 (); foata:sjdm88 (); desainte:lnm85 (); foata_82 (); niukkanen:jpa85 (); chen:mcm03 (); srivastava:ass88 (); hounkonnon:ame00 (), these quantities have not yet been calculated. Here we will compute them for by use of two different approaches.
3.1 Algebraic approach
This functional of the orthogonal Laguerre polynomial can be calculated by use of the linearization formula of Srivastava-Niukkanen srivastava:mcm03 () for the products of various Laguerre polynomials given by
where the coefficients can be expressed as
in terms of the Lauricella’s hypergeometric functions of variables srivastava_85 (). The Pochhammer symbol is . This general relation with the values (, , , , , ) readily yields the following linearization result for the powers of Laguerre polynomials:
for every , . Some examples follow:
For we have that
By considering the definition of the Lauricella function in srivastava_85 (), this expression can be written down as
where we have taken into account that only the term with is different from zero in the previous sum.
For we have
as , and , only the terms with equal to 0 and 1 are different from zero. Then, by considering the number of terms with a given number of indices equal to 1, this expression can be straightforwardly reduced to
which can be further simplified as
3.2 Combinatorial approach
In this approach we begin with the explicit expression of the Laguerre polynomials given by
Recently (sanchezmoreno:jcam09 (); see appendix) it has been found that an integer power of a polynomial can be expressed by use of the multivariate Bell polynomials of Combinatorics riordan_80 (). This result applied to the Laguerre polynomials gives
with for , and the remaining coefficients are given by Eq. (23). Moreover, the Bell polynomials are given by
where the sum runs over all partitions such that
where the parameters are given by Eq. (23), keeping in mind that for every , so that the only non-vanishing terms correspond to those with so that for every .
It is worthwhile to check that for one has that
and that for we have
for the Onicescu information onicescu_craspa66 () of the Laguerre polynomials. Finally, from Eqs. (12) and (26) one has the following alternative expression for the th-order Renyi length of the Laguerre polynomial for ,
which for yields the value
for the Onicescu or second-order Renyi length onicescu_craspa66 () of the Laguerre polynomials.
With these expressions we obtain the same values of and as in the previous subsection:
For we have
Since only the coefficient is different from zero, the previous Bell polynomial contains only one term with index (see Eq. (25)). This yields
in accordance with the result (LABEL:eq_case_n0_algebraic) obtained with the previous method.
The equivalence of Eqs. (20) and (LABEL:eq_renyi_length_bell) for a generic can not be easily proved because of the non-trivial special functions involved.
4 Shannon length: Asymptotics and sharp bounds
The goal of this Section is twofold. First, to study the asymptotics of the Shannon spreading length of the orthonormal Laguerre polynomials and its relation to the standard deviation . Second, to find sharp upper bounds to by use of an information-theoretic optimization procedure.
Although many results have been recently published in the literature (see e.g. igashov:itsf99 (); borwein:sjna08 ()) about the asymptotics of the Laguerre polynomials themselves, they have not yet been successfully used to obtain the asymptotics of functionals of these mathematical functions beyond the -norm method of Aptekarev et al aptekarev:rassm95 (); dehesa:jmp98 (). Here we use the results provided by this method to fix the asymptotics of the Shannon length of these polynomials and its relation to the standard deviation. From Eq. (8) we have that
for the Shannon entropy, and
for the Shannon length of the orthonormal Laguerre polynomials. Where we have used the logarithmic asymptotic behaviour of the digamma function (see Eq. (6.3.18) from abramowitz_64 ()). Moreover, from Eqs. (11) and (32) one finds that
between the asymptotical values of the Shannon length and the standard deviation of the polynomial . It is worth noting that this relation fulfils the general inequality (9) which mutually relates the Shannon length and the standard deviation for general densities. Moreover, the relation (33) for the Rakhmanov densities of Laguerre polynomials is also satisfied by the Rakhmanov densities of Hermite sanchezmoreno:jcam09 (); devicente_04 () and Jacobi guerrero:preprint10 (); devicente_04 () polynomials.
Let us now find sharp upper bounds to the Shannon length by taking into account the non-negativity of the relative Shannon entropy (also called Kullback-Leibler entropy) of two arbitrary probability densities and :
The Jensen inequality implies that . Then, as
we have that the Shannon entropy of is bounded from above by means of
which is normalized to unity. Evaluating now the bound in (LABEL:eq:shannon_bound_f) we obtain
Differentiating this expression with respect to the parameter , and equating it to zero we obtain the value , that is a minimum given the convexity of the previous expression. This yields the bound
for the Shannon length of the Laguerre polynomial . For we have the upper bound
This bound is particularly interesting because it only depends on the expectation value ; the expectation value is, at times, unavailable or difficult to evaluate.
5 Some computational issues
In this section we study various computational issues of the direct spreading measures of Laguerre polynomials. It is worth pointing out that there is no stable numerical algorithm for the computation of the Renyi and Shannon lengths of these polynomials in contrast to the case of orthogonal polynomials on a finite interval for which an efficient algorithm based on the three-term recurrence relation has been recently found by Buyarov et al buyarov:sjsc04 (). Moreover, a naive numerical evaluation of these Laguerre functionals by means of quadratures is not often convenient except for the lowest-order polynomials since the increasing number of integrable singularities spoils any attempt to achieve a reasonable accuracy for arbitrary . Here we carry out the following numerical study. First, we examine the numerical accuracy of the bounds (37) and (38) to the Shannon length of the Laguerre polynomials , fixed, for various degrees by taking into account the optimal values of the parameter in (38) and the optimal values of in (37). These optimal values have been obtained by minimizing the corresponding inequalities numerically. Second, we study the mutual comparison of Fisher, Shannon and Onicescu lengths and the standard deviation of for fixed and various degrees . Finally, we discuss the correlation of the Shannon length and the standard deviation for various pairs , which allow us to find, at times, linear relations between their components.
In Figure 1 it is numerically studied the accuracy of the bounds (37) and (38) to the Shannon length of the Laguerre polynomial given by the optimal values in (38). This is done by comparing the corresponding optimal bounds with the “numerically exact” value of the lengths for the polynomials with degree from to . The graph on the right of the figure gives the relative ratio of the bound given by Eq. (38) with , and the ratio of the bound given by Eq. (37) with and . Notice that the latter bound is always better than the former, as we have the parameter to adjust. The values of optimal pairs and are shown in Tables 1 and 2, respectively. Note that the optimum value is different when considering the bound (37) or (38). Also notice that the optimum values for are , where the density defined in (35) equals the Rakhmanov density for . Remark that the best bounds are obtained for expectation values where is an increasing function of the degree of the polynomial in both cases; this is directly connected with the larger spreading of the polynomial when its degree has higher values.
To study the behaviour of the accuracy of the two previous bounds with respect to , we have done in Figure 2 a study of the Shannon lengths similar to that done in Figure 1 for . The corresponding values and are given in Tables 3 and 4 respectively. The two graphs of the figure show qualitatively similar and quantitatively better results than those found in Figure 1.
In Figures 3 and 4 we study the mutual comparison of various direct spreading measures (namely, the standard deviation and the Fisher, Shannon and the Onicescu or second-order Renyi lengths) of the Laguerre polynomials and , respectively, when the degree varies from to . Several observations are in order. First, all the measures with global character (standard deviation, Shannon and Renyi lengths) grow linearly or quasilinearly when the degree of the polynomial is increasing; essentially because the polynomial spreads more and more. Moreover, they behave so that . Second, the (local) Fisher length decreases when the degree is increasing; essentially, because the polynomial becomes more and more oscillatory, so growing its gradient content. Third, the Fisher length has always a value smaller than all the global spreading measures.
Finally, in Figure 5 we have numerically studied the connection of the Shannon length and the standard deviation of the Laguerre polynomials , with and , when the degree varies from to . This apparent quasilinear behaviour of the Shannon length with respect to the standard deviation is in accordance to the rigorous expression (33), i.e. for .
6 Conclusions and open problems
The global (standard deviation and the Renyi and Shannon lengths) and local (Fisher length) direct spreading measures of the Laguerre polynomials are analytically and numerically studied. Beyond the ordinary moments , and the standard deviation , which have been explicitly given in terms of , we have developed two theoretical approaches of algebraic and combinatorial types to obtain two equivalent analytical expressions for the Renyi lengths of half-integer order. For the Shannon length, whose explicit value is not yet known (in fact, its calculation is a formidable task!), we have found sharp bounds in terms of the expectation value and/or the logarithmic expectation value by means of an information-theoretic-based optimization procedure.
Moreover, the linear correlation of the Shannon length and the standard deviation for the Laguerre polynomials with large degree is underlined. In fact, the correlation factor is not only independent on the parameter but, most importantly, it is the same as for the remaining hypergeometric families on a finite interval (Jacobi polynomials) guerrero:preprint10 (); devicente_04 () or on the whole real line (Hermite polynomials) sanchezmoreno:jcam09 (); devicente_04 ().
Then we carried out a numerical study of the four direct spreading measures of Laguerre polynomials. Let us remark, among other results, that the Fisher length has the smallest value, and the Shannon length depends quasilinearly on the standard deviation.
Finally, let us highlight a number of open information-theoretic problems related to Laguerre polynomials: (i) to find the asymptotics of the entropic moments and, subsequently, the Renyi lengths in the spirit of aptekarev:rassm95 (); dehesa:jmp98 (), (ii) to identify the most general class of polynomials for which the asymptotical relation (33) of the Shannon length and the standard deviation is fulfilled, and (iii) to characterize the most general class of polynomials for which the ratio between these two direct spreading measures is a constant (i.e., it does not depend on the degree nor the parameters of the polynomials) as already pointed out in devicente_04 ().
We are very grateful to Junta de Andalucía for the grants FQM-2445 and FQM-4643, and the Ministerio de Ciencia e Innovación for the grant FIS2008-02380. We belong to the research group FQM-207. Daniel Manzano acknowledges the fellowship BES-2006-13234.
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