# On recurrence relations for radial wave functions for the N-th dimensional oscillators and hydrogenlike atoms: analytical and nuberical study.

Abstract. Using a general procedure for finding recurrence
relations for hypergeometric functions and polynomials introduced by
Cardoso et al. [J. Phys. A, 36 (2003), pp. 2055-2068] we obtain some new
recurrence relations for the radial wave functions of the N-th
dimensional isotropic harmonic oscillators as well as the hydrogenlike
atoms. A numerical analysis of such recurrences is also presented.

Key words. wave functions, linear recurrence relations, Laguerre polynomials

AMS subject classifications. 33C45, 111337, 81V45, 81Q99

1. Introduction. There are many applications in modern physics that require the knowledge of the wave functions of hydrogenlike atoms and isotropic harmonic oscillators (see e.g. [13, 16] and references therein). Of particular interest is the numerical implementation of such functions. The most natural, efficient and fast way for generating such functions is to use recurrence relations. In fact, although the explicit expressions are known, their numerical implementations are usually very unstable.

In this paper we will continue the research started in [8] for obtaining recurrence relations and ladder-type operators for the N-th dimensional isotropic harmonic oscillators and the hydrogenlike atoms. In fact, using the technique of [8] we will obtain some new recurrences on one hand, and on the other we will present a comparative numerical analysis of the obtained recurrence relations for generating numerically the corresponding eigen-functions. The numerical analysis of the linear recurrences, i.e., the rounding errors bounds, stability of the scheme, etc. is, in general, very complicate (see the nice paper [5] and references therein). So we will restrict ourselves to the discussion of the numerical examples. Even in this framework of no rigorous numerical analysis it is possible to conclude what kind of relations are more useful for numerical computations and what are not. This is the main original contribution of the present paper. Finally, let us mention that the present method for finding recurrence relations can be extended to any quantum system whose (radial) wave function are proportional to hypergeometric-type functions (see e.g. [4]).

The structure of the paper is as follows: In sections 2 and 3 the isotropic oscillator and hydrogenlike atoms radial wave functions are introduced. In both cases we include some new recurrence relations as well as the detailed discussion of their computational applications. Finally, an appendix with the required formulas of the Laguerre polynomials is included.

2. The isotropic harmonic oscillator. The N-dimensional isotropic harmonic oscillator (I.H.O.) is described by the Schrodinger equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Its solutions are of the form [PSI] = [R.sup.(N).sub.nl] (r) [Y.sub.lm] ([[OMEGA].sub.N]), where [R.sup.(N).sub.nl] (r) is the radial part, usually called the radial wave functions, defined by (see e.g. [3, 6])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

being n = 0, 1, 2.... and l = 0, 1, 2,.. . , the quantum numbers, and N [greater than or equal to] 3 the dimension of the space. The angular part [Y.sub.lm]([[OMEGA].sub.N]) are the so-called Nth-spherical or hyperspherical harmonics [3, 14]. In the following, we will assume that the parameters n, l, N are nonnegative integers.

2.1. Recurrence relations for the I.H.O. radial functions. For the functions (2.1) the following theorem hold [8]

THEOREM 2.1. Let [R.sup.(N).sub.nl] (r), [R.sup.(N).sub.n+n1,l+l1] (r) and [R.sup.(N).sub.n+n2,l+l2] (r) be three different radial functions of the N-th dimensional isotropic harmonic oscillator, where nl , n2 and 11 , 12 are integers such that

min (n+n1,n+n2,l+l1+l+l2) [greater than or equal to] 0.

Then, there exist three all non vanishing polynomials in r, [A.sub.o], [A.sub.1], and [A.sub.2], such that

The proof of this theorem can be found in [8]. A key point on the proof was the recurrence relation,

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where s = [lambda][r.sup.2], and Ci(s), i = 0, 1, 2, are not all three vanishing polynomials. Moreover, in [8] it was shown that

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From the above theorem and the relation (2.3) it is very simple to obtain a la carte relations between three different radial functions of the I.H.O. Let us point out that, in general, it is not easy to obtain the coefficients Ci in (2.3), nevertheless, combining in a certain way the known relations of the Laguerre polynomials (see the appendix A) they can be found. This has been shown in [7, 8]. We will show here how this works in one of the new examples, for the others the computations are similar. Here we present some examples. The first four can be found in [8] and the other five seem to be new.

* n1 = -1, n2 = 1, 11 = 12 = 0 [8, page 2059]

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(2.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(2.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

we are looking for a relation of type (2.2)

For finding the polynomials Ai, i = 0, 1, 2 we use (2.4) where Ci, i = 0, 1, 2 are the polynomials in (2.3). Putting a = l + N/2 - 1 we have

Now we substitute the functions [L.sup.a+1.sub.n-1] (s) and [L.sup.a-i.sub.n+1] (s) using the relations (A.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

respectively. This leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Comparing the last formula with the three-term recurrence relation TTRR (A.5) for the Laguerre polynomials we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and therefore for the wave functions we find

(2.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Analogously, we can find the following three recurrence relations

(2.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

2.2. Ladder-type relations for the I.H.O. radial functions. Another important relations for the functions (2.1) are the so-called ladder operators. The following result was proved in [8].

THEOREM 2.2. Let [R.sup.(N).sub.n,l] (r) and [R.sup.(N).sub.n+n1,l+li (r) be two radial functions of the N-th dimensional isotropic harmonic oscillator and let min (n + n1, l + l1) [greater than or equal to] 0 and [([n.sub.1).sup.2] + [(l1).sup.2] [not equal to] 0, where [n.sub.1] and [l.sub.1] are integers. Then, there exist three not all vanishing polynomials in r, [A.sub.O], [A.sub.1], and [A.sub.2], such that

(2.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The proof of this theorem can be found in [8]. A key point on the proof was the recurrence relation,

(2.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [B.sub.o], [B.sub.1], and [B.sub.2] are not all three vanishing polynomials. Moreover, (2.13) can be rewritten as

(2.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

To obtain the unknown polynomials [B.sub.0], [B.sub.1], and [B.sub.2] in (2.14) we can proceed as in the previous section, i.e., use the Eqs. (A.1)-(A.5) to transform (2.14) into one of the formulas (A.1)-(A.5) or in a sum of linearly independent Laguerre polynomials and solve the resulting equations for the unknown coefficients. Let us consider some examples. The first two are taken from [8] and the other two are new.

* n1 = 0, l1 = 1 [8, page 2061]

(2.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

* n1 = 2, l1 = 0 [8, page 2061]

(2.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

* [n.sub.1] = 2, [l.sub.1] = 0 Introducing these values into (2.14) and putting a = n + N/2 - 1 we get

(2.18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now, using relations (A.2) and the TTRR (A.5), (2.18) becomes into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Comparing with the TTRR (A.5) we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and therefore, for the wave functions we find (see (2.15))

* [n.sub.1] = 0, l1 = 2 Following the same technique and using now twice, in the resulting (2.14), formulas (A.3) and (A.2) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

2.3. Numerical analysis of the recurrences. In this section we will present the numerical analysis of some of the recurrence relations and ladder-type operators for the I.H.O. We have used the commercial program MATLAB [12].

First of all, let us point out that if we choose the values of n and l large enough and compute the radial wave functions [R.sup.(N).sub.n,l] (r) using the definition (2.1) we observe a picture of an unbounded function that do not corresponds to the real ones. This picture is probably due to numerical instabilities and consequently, we can not directly evaluate these functions for a given r by means of (2.1). In figure 2.1 we represent the maximum value of the parameters n, l for which the direct computation works, i.e., the region below the curve defined by these points is the region of computational validity of (2.1). Therefore, if we want to have the values of the radial functions for n, l large enough, it is necessary to apply the recurrence relations (RR) in order to calculate them.

From the previous sections it follows that the recurrence relations of the I.H.O. can be classify in three different types:

* RRs which involve functions Rn;,h with indices (n,l), (n [+ or -] 1, l) (see (2.5)); (n,l), (n, l [+ or -] 1) (see (2.6)); (n, l) and (n [+ or -] 1, l [+ or -] 1) (see (2.9)); and (n, l) and (n [+ or -] 1, l [+ or -] 1) (see (2.10)). We will call them the regular recurrences.

* The other RRs, e.g. the ones which involve the indices (n, l), (n, l - 1), (n + 1, l); (n,l), (n, l + 1), (n - 1,l); (n, l), (n,l + 1), (n + 2, l); (n, l), (n, l + 1), (n - 2, l); etc. (see e.g. (2.7), (2.8), (2.11), (2.12)).

* Relations which involve the derivatives, i.e., ladder-type relations.

[FIGURE 2.1 OMITTED]

In this section we evaluate the radial wave functions by using different recurrence relations with the aim to discuss their effectiveness (based on the time of the numerical simulations and convergence of the formulas). In all the numerical computations of [R.sub.n,l] (r) we use [lambda] = 1 and N=3.

[FIGURE 2.2 OMITTED]

First of all, we evaluate the function [R.sub.l] (r) in 1000 points, corresponding to the elements of the vector r = (4.01, 4.02, ... ,14) (in the matlab notation r = [4.01 : 0.01 : 14]), by using the formulas (2.5), (2.6) and (2.10). In figure 2.2 we compare the computational time to do these operations in each case when n = l (notice that in this case we need to apply the corresponding RR n - 1 times).

[FIGURE 2.3 OMITTED]

We observe that formula (2.5) allows us to compute these functions faster. The difference in time among these methods is more appreciable when n is bigger. We would like to remark that in order to apply the formula (2.5) to obtain [R.sub.n,l] (r), we fix l and change the first argu ment from 1 to n - 1. In addition, we need to know the initial conditions of the RR, which in this case are [R.sub.0,1] (r) and [R.sub.1,1] (r). Both functions are related with the Laguerre polynomials [L.sup.0.sub.l+N/2-1](x) = land [L.sup.l+N/2-1].sub.1](x) = x, respectively, regardless of the value of l. Furthermore, for using the relation (2.6) one should use the initial conditions [R.sub.n,o] (r) and [R.sub.n,1](r), but they can not be calculated by formula (2.1) for large values of n (see figure 2.1) as already pointed out. So, the initial conditions should be calculated using the recurrence formula (2.5). Moreover, when we use the relation (2.6) we see that there exists an interval close to r = 0 where [R.sub.n,l] (r) diverges (oscillates without any pattern) and this interval becomes larger when n increases. This divergence, at the vicinity of zero, also appears when we use (2.10). In the last case, one should also use (2.5) to obtain the initial condition.

The recurrences (2.7), (2.8), (2.11) and (2.12) are even more complicated to use than (2.6) and (2.10). They should be used together with (2.5), not only because the initial conditions, but also because their own nature (they mix both indices). Let us check if [R.sub.n,l](r) can be obtained faster when we combine for example (2.11) with (2.5), instead of using (2.5) alone. In figure 2.3 we show the computer time to obtain [R.sub.n,il (r) versus n (and fixing l = 40) for the relation (2.5) alone, and combining (2.11) and (2.5) together. Both functions are computed in 1000 points corresponding to the elements of the vector [0.01 : 0.01 : 10]. From this figure we conclude that the formula (2.5) is the best one for computing [R.sub.n,l](r).

Concerning the ladder-type relations, let us mention that the relations (2.16) and (2.17) behave numerically unstable for large values of l and n, respectively. Therefore, they are not useful to compute numerically the radial wave functions but instead of this we can use them, together with (2.5) for finding the derivative of the radial wave functions (see figure 2.4).

[FIGURE 2.4 OMITTED]

3. Radial functions for the Hydrogen atom. In this section we will provide a similar study for the Hydrogen atom described by the Schrodinger equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The solution is given by [PSI] = [R.sup.(N).sub.nl] (r) [Y.sub.lm], ([OMEGA]N), where the radial part [R.sup.(N).sup.nl] (r) is defined by [2, 9]

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here n = l + 1,l + 2,... and l = 0, 1, 2,... are the quantum numbers, N [greater than or equal to] 3 is the dimension of the space, and the normalizing constant [N.sup.(N).sub.n,l] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

As we already pointed out in [8], the Laguerre polynomials that appear in the expression of the radial functions are not the classical ones [L.sup.a.sub.n] (x) in the sense that the parameter a as well as the variable x depend on the degree of the polynomials, n. When the parameters of the classical polynomials depend on n, the polynomials are orthogonal with respect to a variant weights (for more details see e.g. [10, 11, 18]). Nevertheless, as we showed in [8], using the algebraic properties of the classical Laguerre polynomials (A.1)-(A.5) one can derive the algebraic relations of the radial wave functions.

3.1. Recurrence relations and ladder-type operators for the radial functions. In [8] we proved the following

THEOREM 3.1. Let the functions [R.sup.(N).sub.nl] [(n+N-3/2)r], [R.sup.(N).sub.n+n1,l+l1] [(n+n1+N-3/2)r] and [R.sup.(N).sub.n2,l+l2 [(n + n2 + N-3/2) r] be three different radial functions of the N-th Hydrogen atom and [n.sub.1], [n.sub.2] and [l.sub.1],[l.sub.2] integers such that

min (n+n1,n+n2,l+1,l+l2) [greater than or equal to] 0.

Then, there exist not all three vanishing polynomials in r, [A.sub.0], [A.sub.1], and [A.sub.2], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Moreover, the expressions for the polynomial coefficients in the above formula are given by [8, Eq. (4.7) page 2063]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [A.sup.*.sub.0], [A.sup.*.sub.1], and [A.sup.*.sub.2], are the non all three vanishing polynomials of the linear relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and a = 2l + N - 2, m = n - l -1

As examples of recurrences obtained by using the Theorem 3.1 we have the following (the first two are taken from [8], the other five are seem to be new). Other cases can be obtained in the same way.

(3.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3.2. Ladder-type relations for the radial functions of the Hydrogen atom. Our starting point is the following

THEOREM 3.2. Let [R.sup.(N).sub.nl] [(n + [N-3/2]) r], and [R.sup.(N).sub.n+n1,l+l1] [(n + [n.sub.1] + [N-3/2) r] two different radial functions of the Hydrogen atom and d/dr [R.sup.(N).sub.nl] [(n + N-3/2) r], the first derivative with respect to r, where nl and 11 are integers such that min (n + nl, l + 11) [less than or equal to] 0, ([n.sub.1])2 + [(l1).sup.2] [equal to] 0. Then, there exist not all three vanishing polynomials in r, Ao, [A.sub.1] and [A.sub.2], such that

(3.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The proof is similar to the one of Theorem 2.2 and was given in [8]. In fact in this case the relation (3.7) becomes into (see [8, (4.13) page 2064])

(3.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the polynomials [B.sub.0], [B.sub.1], and [B.sub.2] are such that

(3.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Again, from the above theorem it is easy to obtain several relations for the radial wave functions of the Hydrogen atom and, in particular, the ladder operators in n and l, respectively. We will present here some of them. The first three are taken from [8] and the other two are seem to be new.

(3.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now, using three times relations (A.4) and (A.2), (3.13) becomes into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Comparing with the TTRR (A.5) we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and therefore (see (3.8)), for the wave functions we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3.3. Numerical analysis of the recurrences. As in the previous case, the radial wave functions for n and l large enough, calculated by using the functions (3.1) (by means of the Laguerre polynomials), behave as an unbounded function, and therefore we can not directly evaluate these functions for a given r. In figure 3.1 we show the critical values of n,,,,,, for a given l for which the radial functions can be computed using the explicit expression (3.1). The region where the explicit formula works is the one defined by the two curves in figure 3.1, (it is defined the values l + 1 [less than or equal to] n [less than or equal to] [n.sub.max]).

[FIGURE 3.1 OMITTED]

Again we have studied numerically how to obtain the radial wave functions applying the recurrence relations (RR) of two types: the first ones are represented by the formulas (3.2), (3.3) and (3.4) and the second ones correspond to the ladder-type RRs (3.10), (3.11) and (3.12). For the former RRs we observe that

* the relation (3.2) is the most useful one because the initial conditions Ri+l,c and [Rl+2,l] can be calculated by using the Laguerre polynomials (notice that in this case we fix l and start with n = l + 2).

* in the relation (3.3) we fix n and increase the value of l in each step up to its maximum value. From figure 3.1 we conclude that for large value of n, we can not use the initial condition calculated by using Laguerre polynomials (formula (3.1)) and instead of this we should use the RR (3.2). For this reason, this relation can not be used alone (the same is true for the RR (3.4)). In addition, notice that the initial conditions are two radial functions, [R.sub.n,l-1] and [R.sub.n,l] evaluated at r, whereas (3.2) gives us the same functions but at nr (this implies a rescaling of the argument).

* From figure 3.2 we observe that the computational time is less when we use the formula (3.2). Moreover, the formulas (3.3) and (3.4) are numerically unstable at the vicinity of zero and this region becomes larger when n and l increases.

* As in the case of the I.H.O. the higher order relation (3.5) and (3.6) are not useful for numerical evaluation of the Radial functions. For the ladder-type relations we would like to point out the following remarks

* Notice that the expression (3.12) involves two wave functions evaluated at different points, so in each step to calculate [R.sub.n+l,l] [(n + 1)r/n] we need to know [R.sub.n,l](r), but in the previous step we calculate [R.sub.n,l] at nr/(n - 1), not at r. For this reason this RRs it is not useful for obtaining numerically the radial wave functions.

* As in the previous case, we can use the ladder-type relations (3.10), (3.12) and (3.11) together to (3.2), to compute the derivative of the radial wave function. As an example see the figure 3.3, where we plot the radial wave function and its derivative by using (3.10) for n = 60 and l = 50.

[FIGURE 3.2 OMITTED]

In general we verify that the ladder-type RRs are not useful in order to compute numerically the radial wave functions. Nevertheless they can be used, together with (3.2) for finding the derivative of the radial wave functions.

[FIGURE 3.3 OMITTED]

Programs. For the numerical simulations presented here we have used the commercial program MATLAB. The used source code can be obtained by request via e-mail to niurka@euler.us.esorran@us.es.

Acknowledgements. The authors thank J. Petronilho for interesting discussions as well as the unknown referee for his comments that help us to improve the paper. This research has been supported by the Programa de Acciones Integradas Hispano-Lusas HP2002-065 & E-6/03. The authors were also partially supported by the the DGES grant BFM 2003-06335C03-01 and PAI grant FQM-0262 (RAN); CMUC (JLC); and the DGES grant FIS2005-973 and PAI grant FQM-0207 (NRQ).

Appendix. The Laguerre polynomials. The Laguerre polynomials Ln' defined by the hypergeometric series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

These polynomials satisfy the following useful recurrence and differential-recurrence relations (see e.g. [1, 15, 17])

(A.1) d/dx [L.sup.a.sub.n](x) = [L.sub.a+1][n-x](x),

(A.2) x[L.sup.a+1.sub.n] (x) = (n + a + 1) [L.sup.a.sub.n] (x) - (n + 1) [L.sup.a.sub.n+1](x),

(A.3) x[L.sup.a+1.sub.n] (x) = (n + a) [L.sup.a.sub.n-1] (x) - (n - x) [L.sup.a.sub.n(x),

(A.4) [L.sup.a-1.sub.n] (x) = [L.sup.a.sub.n] (x) - [[L.sup.a.sub.n-1](x)

(A.5) (n + 1) [L.sub.a.sub.n+1](x) - (2n + a + 1 - x)[L.sup.a.sub.n] (x) + (n + a) [L.sup.a.sub.n-1](x) = 0.

* Received March 11, 2004. Accepted for publication June 3, 2005. Recommended by F. Marcellon.

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[17] G. SZEG6, Orthogonal Polynomials, Amer. Math. Soc. Coll. Pub., 23, American Mathematical Society, Providence, Rhode Island, 1975.

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R. ALVAREZ-NODARSE ([dagger]), J. L. CARDOSO ([double-dagger]), AND N. R. QUINTERO ([section])

([dagger]) Departamento de Anolisis Matemotico, Facultad de Matemotica, Universidad de Sevilla. Apdo. Postal 1160, Sevilla, E-41080, Sevilla, Spain (ran@us.es).

([double-dagger]) Departamento de Matematica, Universidade de Tras-os-Montes e Alto Douro. Apartado 202, 5001 - 911 Vila Real, Portugal (j1uis@utad.pt).

([section]) Dpto. Fisica Aplicada I, EUP, Universidad de Sevilla, c/Virgen de Africa 7, 41011, Sevilla, Spain (niurka@us.es).

Key words. wave functions, linear recurrence relations, Laguerre polynomials

AMS subject classifications. 33C45, 111337, 81V45, 81Q99

1. Introduction. There are many applications in modern physics that require the knowledge of the wave functions of hydrogenlike atoms and isotropic harmonic oscillators (see e.g. [13, 16] and references therein). Of particular interest is the numerical implementation of such functions. The most natural, efficient and fast way for generating such functions is to use recurrence relations. In fact, although the explicit expressions are known, their numerical implementations are usually very unstable.

In this paper we will continue the research started in [8] for obtaining recurrence relations and ladder-type operators for the N-th dimensional isotropic harmonic oscillators and the hydrogenlike atoms. In fact, using the technique of [8] we will obtain some new recurrences on one hand, and on the other we will present a comparative numerical analysis of the obtained recurrence relations for generating numerically the corresponding eigen-functions. The numerical analysis of the linear recurrences, i.e., the rounding errors bounds, stability of the scheme, etc. is, in general, very complicate (see the nice paper [5] and references therein). So we will restrict ourselves to the discussion of the numerical examples. Even in this framework of no rigorous numerical analysis it is possible to conclude what kind of relations are more useful for numerical computations and what are not. This is the main original contribution of the present paper. Finally, let us mention that the present method for finding recurrence relations can be extended to any quantum system whose (radial) wave function are proportional to hypergeometric-type functions (see e.g. [4]).

The structure of the paper is as follows: In sections 2 and 3 the isotropic oscillator and hydrogenlike atoms radial wave functions are introduced. In both cases we include some new recurrence relations as well as the detailed discussion of their computational applications. Finally, an appendix with the required formulas of the Laguerre polynomials is included.

2. The isotropic harmonic oscillator. The N-dimensional isotropic harmonic oscillator (I.H.O.) is described by the Schrodinger equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Its solutions are of the form [PSI] = [R.sup.(N).sub.nl] (r) [Y.sub.lm] ([[OMEGA].sub.N]), where [R.sup.(N).sub.nl] (r) is the radial part, usually called the radial wave functions, defined by (see e.g. [3, 6])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

being n = 0, 1, 2.... and l = 0, 1, 2,.. . , the quantum numbers, and N [greater than or equal to] 3 the dimension of the space. The angular part [Y.sub.lm]([[OMEGA].sub.N]) are the so-called Nth-spherical or hyperspherical harmonics [3, 14]. In the following, we will assume that the parameters n, l, N are nonnegative integers.

2.1. Recurrence relations for the I.H.O. radial functions. For the functions (2.1) the following theorem hold [8]

THEOREM 2.1. Let [R.sup.(N).sub.nl] (r), [R.sup.(N).sub.n+n1,l+l1] (r) and [R.sup.(N).sub.n+n2,l+l2] (r) be three different radial functions of the N-th dimensional isotropic harmonic oscillator, where nl , n2 and 11 , 12 are integers such that

min (n+n1,n+n2,l+l1+l+l2) [greater than or equal to] 0.

Then, there exist three all non vanishing polynomials in r, [A.sub.o], [A.sub.1], and [A.sub.2], such that

The proof of this theorem can be found in [8]. A key point on the proof was the recurrence relation,

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where s = [lambda][r.sup.2], and Ci(s), i = 0, 1, 2, are not all three vanishing polynomials. Moreover, in [8] it was shown that

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From the above theorem and the relation (2.3) it is very simple to obtain a la carte relations between three different radial functions of the I.H.O. Let us point out that, in general, it is not easy to obtain the coefficients Ci in (2.3), nevertheless, combining in a certain way the known relations of the Laguerre polynomials (see the appendix A) they can be found. This has been shown in [7, 8]. We will show here how this works in one of the new examples, for the others the computations are similar. Here we present some examples. The first four can be found in [8] and the other five seem to be new.

* n1 = -1, n2 = 1, 11 = 12 = 0 [8, page 2059]

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(2.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(2.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

we are looking for a relation of type (2.2)

For finding the polynomials Ai, i = 0, 1, 2 we use (2.4) where Ci, i = 0, 1, 2 are the polynomials in (2.3). Putting a = l + N/2 - 1 we have

Now we substitute the functions [L.sup.a+1.sub.n-1] (s) and [L.sup.a-i.sub.n+1] (s) using the relations (A.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

respectively. This leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Comparing the last formula with the three-term recurrence relation TTRR (A.5) for the Laguerre polynomials we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and therefore for the wave functions we find

(2.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Analogously, we can find the following three recurrence relations

(2.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

2.2. Ladder-type relations for the I.H.O. radial functions. Another important relations for the functions (2.1) are the so-called ladder operators. The following result was proved in [8].

THEOREM 2.2. Let [R.sup.(N).sub.n,l] (r) and [R.sup.(N).sub.n+n1,l+li (r) be two radial functions of the N-th dimensional isotropic harmonic oscillator and let min (n + n1, l + l1) [greater than or equal to] 0 and [([n.sub.1).sup.2] + [(l1).sup.2] [not equal to] 0, where [n.sub.1] and [l.sub.1] are integers. Then, there exist three not all vanishing polynomials in r, [A.sub.O], [A.sub.1], and [A.sub.2], such that

(2.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The proof of this theorem can be found in [8]. A key point on the proof was the recurrence relation,

(2.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [B.sub.o], [B.sub.1], and [B.sub.2] are not all three vanishing polynomials. Moreover, (2.13) can be rewritten as

(2.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

To obtain the unknown polynomials [B.sub.0], [B.sub.1], and [B.sub.2] in (2.14) we can proceed as in the previous section, i.e., use the Eqs. (A.1)-(A.5) to transform (2.14) into one of the formulas (A.1)-(A.5) or in a sum of linearly independent Laguerre polynomials and solve the resulting equations for the unknown coefficients. Let us consider some examples. The first two are taken from [8] and the other two are new.

* n1 = 0, l1 = 1 [8, page 2061]

(2.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

* n1 = 2, l1 = 0 [8, page 2061]

(2.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

* [n.sub.1] = 2, [l.sub.1] = 0 Introducing these values into (2.14) and putting a = n + N/2 - 1 we get

(2.18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now, using relations (A.2) and the TTRR (A.5), (2.18) becomes into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Comparing with the TTRR (A.5) we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and therefore, for the wave functions we find (see (2.15))

* [n.sub.1] = 0, l1 = 2 Following the same technique and using now twice, in the resulting (2.14), formulas (A.3) and (A.2) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

2.3. Numerical analysis of the recurrences. In this section we will present the numerical analysis of some of the recurrence relations and ladder-type operators for the I.H.O. We have used the commercial program MATLAB [12].

First of all, let us point out that if we choose the values of n and l large enough and compute the radial wave functions [R.sup.(N).sub.n,l] (r) using the definition (2.1) we observe a picture of an unbounded function that do not corresponds to the real ones. This picture is probably due to numerical instabilities and consequently, we can not directly evaluate these functions for a given r by means of (2.1). In figure 2.1 we represent the maximum value of the parameters n, l for which the direct computation works, i.e., the region below the curve defined by these points is the region of computational validity of (2.1). Therefore, if we want to have the values of the radial functions for n, l large enough, it is necessary to apply the recurrence relations (RR) in order to calculate them.

From the previous sections it follows that the recurrence relations of the I.H.O. can be classify in three different types:

* RRs which involve functions Rn;,h with indices (n,l), (n [+ or -] 1, l) (see (2.5)); (n,l), (n, l [+ or -] 1) (see (2.6)); (n, l) and (n [+ or -] 1, l [+ or -] 1) (see (2.9)); and (n, l) and (n [+ or -] 1, l [+ or -] 1) (see (2.10)). We will call them the regular recurrences.

* The other RRs, e.g. the ones which involve the indices (n, l), (n, l - 1), (n + 1, l); (n,l), (n, l + 1), (n - 1,l); (n, l), (n,l + 1), (n + 2, l); (n, l), (n, l + 1), (n - 2, l); etc. (see e.g. (2.7), (2.8), (2.11), (2.12)).

* Relations which involve the derivatives, i.e., ladder-type relations.

[FIGURE 2.1 OMITTED]

In this section we evaluate the radial wave functions by using different recurrence relations with the aim to discuss their effectiveness (based on the time of the numerical simulations and convergence of the formulas). In all the numerical computations of [R.sub.n,l] (r) we use [lambda] = 1 and N=3.

[FIGURE 2.2 OMITTED]

First of all, we evaluate the function [R.sub.l] (r) in 1000 points, corresponding to the elements of the vector r = (4.01, 4.02, ... ,14) (in the matlab notation r = [4.01 : 0.01 : 14]), by using the formulas (2.5), (2.6) and (2.10). In figure 2.2 we compare the computational time to do these operations in each case when n = l (notice that in this case we need to apply the corresponding RR n - 1 times).

[FIGURE 2.3 OMITTED]

We observe that formula (2.5) allows us to compute these functions faster. The difference in time among these methods is more appreciable when n is bigger. We would like to remark that in order to apply the formula (2.5) to obtain [R.sub.n,l] (r), we fix l and change the first argu ment from 1 to n - 1. In addition, we need to know the initial conditions of the RR, which in this case are [R.sub.0,1] (r) and [R.sub.1,1] (r). Both functions are related with the Laguerre polynomials [L.sup.0.sub.l+N/2-1](x) = land [L.sup.l+N/2-1].sub.1](x) = x, respectively, regardless of the value of l. Furthermore, for using the relation (2.6) one should use the initial conditions [R.sub.n,o] (r) and [R.sub.n,1](r), but they can not be calculated by formula (2.1) for large values of n (see figure 2.1) as already pointed out. So, the initial conditions should be calculated using the recurrence formula (2.5). Moreover, when we use the relation (2.6) we see that there exists an interval close to r = 0 where [R.sub.n,l] (r) diverges (oscillates without any pattern) and this interval becomes larger when n increases. This divergence, at the vicinity of zero, also appears when we use (2.10). In the last case, one should also use (2.5) to obtain the initial condition.

The recurrences (2.7), (2.8), (2.11) and (2.12) are even more complicated to use than (2.6) and (2.10). They should be used together with (2.5), not only because the initial conditions, but also because their own nature (they mix both indices). Let us check if [R.sub.n,l](r) can be obtained faster when we combine for example (2.11) with (2.5), instead of using (2.5) alone. In figure 2.3 we show the computer time to obtain [R.sub.n,il (r) versus n (and fixing l = 40) for the relation (2.5) alone, and combining (2.11) and (2.5) together. Both functions are computed in 1000 points corresponding to the elements of the vector [0.01 : 0.01 : 10]. From this figure we conclude that the formula (2.5) is the best one for computing [R.sub.n,l](r).

Concerning the ladder-type relations, let us mention that the relations (2.16) and (2.17) behave numerically unstable for large values of l and n, respectively. Therefore, they are not useful to compute numerically the radial wave functions but instead of this we can use them, together with (2.5) for finding the derivative of the radial wave functions (see figure 2.4).

[FIGURE 2.4 OMITTED]

3. Radial functions for the Hydrogen atom. In this section we will provide a similar study for the Hydrogen atom described by the Schrodinger equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The solution is given by [PSI] = [R.sup.(N).sub.nl] (r) [Y.sub.lm], ([OMEGA]N), where the radial part [R.sup.(N).sup.nl] (r) is defined by [2, 9]

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here n = l + 1,l + 2,... and l = 0, 1, 2,... are the quantum numbers, N [greater than or equal to] 3 is the dimension of the space, and the normalizing constant [N.sup.(N).sub.n,l] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

As we already pointed out in [8], the Laguerre polynomials that appear in the expression of the radial functions are not the classical ones [L.sup.a.sub.n] (x) in the sense that the parameter a as well as the variable x depend on the degree of the polynomials, n. When the parameters of the classical polynomials depend on n, the polynomials are orthogonal with respect to a variant weights (for more details see e.g. [10, 11, 18]). Nevertheless, as we showed in [8], using the algebraic properties of the classical Laguerre polynomials (A.1)-(A.5) one can derive the algebraic relations of the radial wave functions.

3.1. Recurrence relations and ladder-type operators for the radial functions. In [8] we proved the following

THEOREM 3.1. Let the functions [R.sup.(N).sub.nl] [(n+N-3/2)r], [R.sup.(N).sub.n+n1,l+l1] [(n+n1+N-3/2)r] and [R.sup.(N).sub.n2,l+l2 [(n + n2 + N-3/2) r] be three different radial functions of the N-th Hydrogen atom and [n.sub.1], [n.sub.2] and [l.sub.1],[l.sub.2] integers such that

min (n+n1,n+n2,l+1,l+l2) [greater than or equal to] 0.

Then, there exist not all three vanishing polynomials in r, [A.sub.0], [A.sub.1], and [A.sub.2], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Moreover, the expressions for the polynomial coefficients in the above formula are given by [8, Eq. (4.7) page 2063]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [A.sup.*.sub.0], [A.sup.*.sub.1], and [A.sup.*.sub.2], are the non all three vanishing polynomials of the linear relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and a = 2l + N - 2, m = n - l -1

As examples of recurrences obtained by using the Theorem 3.1 we have the following (the first two are taken from [8], the other five are seem to be new). Other cases can be obtained in the same way.

(3.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3.2. Ladder-type relations for the radial functions of the Hydrogen atom. Our starting point is the following

THEOREM 3.2. Let [R.sup.(N).sub.nl] [(n + [N-3/2]) r], and [R.sup.(N).sub.n+n1,l+l1] [(n + [n.sub.1] + [N-3/2) r] two different radial functions of the Hydrogen atom and d/dr [R.sup.(N).sub.nl] [(n + N-3/2) r], the first derivative with respect to r, where nl and 11 are integers such that min (n + nl, l + 11) [less than or equal to] 0, ([n.sub.1])2 + [(l1).sup.2] [equal to] 0. Then, there exist not all three vanishing polynomials in r, Ao, [A.sub.1] and [A.sub.2], such that

(3.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The proof is similar to the one of Theorem 2.2 and was given in [8]. In fact in this case the relation (3.7) becomes into (see [8, (4.13) page 2064])

(3.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the polynomials [B.sub.0], [B.sub.1], and [B.sub.2] are such that

(3.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Again, from the above theorem it is easy to obtain several relations for the radial wave functions of the Hydrogen atom and, in particular, the ladder operators in n and l, respectively. We will present here some of them. The first three are taken from [8] and the other two are seem to be new.

(3.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now, using three times relations (A.4) and (A.2), (3.13) becomes into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Comparing with the TTRR (A.5) we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and therefore (see (3.8)), for the wave functions we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3.3. Numerical analysis of the recurrences. As in the previous case, the radial wave functions for n and l large enough, calculated by using the functions (3.1) (by means of the Laguerre polynomials), behave as an unbounded function, and therefore we can not directly evaluate these functions for a given r. In figure 3.1 we show the critical values of n,,,,,, for a given l for which the radial functions can be computed using the explicit expression (3.1). The region where the explicit formula works is the one defined by the two curves in figure 3.1, (it is defined the values l + 1 [less than or equal to] n [less than or equal to] [n.sub.max]).

[FIGURE 3.1 OMITTED]

Again we have studied numerically how to obtain the radial wave functions applying the recurrence relations (RR) of two types: the first ones are represented by the formulas (3.2), (3.3) and (3.4) and the second ones correspond to the ladder-type RRs (3.10), (3.11) and (3.12). For the former RRs we observe that

* the relation (3.2) is the most useful one because the initial conditions Ri+l,c and [Rl+2,l] can be calculated by using the Laguerre polynomials (notice that in this case we fix l and start with n = l + 2).

* in the relation (3.3) we fix n and increase the value of l in each step up to its maximum value. From figure 3.1 we conclude that for large value of n, we can not use the initial condition calculated by using Laguerre polynomials (formula (3.1)) and instead of this we should use the RR (3.2). For this reason, this relation can not be used alone (the same is true for the RR (3.4)). In addition, notice that the initial conditions are two radial functions, [R.sub.n,l-1] and [R.sub.n,l] evaluated at r, whereas (3.2) gives us the same functions but at nr (this implies a rescaling of the argument).

* From figure 3.2 we observe that the computational time is less when we use the formula (3.2). Moreover, the formulas (3.3) and (3.4) are numerically unstable at the vicinity of zero and this region becomes larger when n and l increases.

* As in the case of the I.H.O. the higher order relation (3.5) and (3.6) are not useful for numerical evaluation of the Radial functions. For the ladder-type relations we would like to point out the following remarks

* Notice that the expression (3.12) involves two wave functions evaluated at different points, so in each step to calculate [R.sub.n+l,l] [(n + 1)r/n] we need to know [R.sub.n,l](r), but in the previous step we calculate [R.sub.n,l] at nr/(n - 1), not at r. For this reason this RRs it is not useful for obtaining numerically the radial wave functions.

* As in the previous case, we can use the ladder-type relations (3.10), (3.12) and (3.11) together to (3.2), to compute the derivative of the radial wave function. As an example see the figure 3.3, where we plot the radial wave function and its derivative by using (3.10) for n = 60 and l = 50.

[FIGURE 3.2 OMITTED]

In general we verify that the ladder-type RRs are not useful in order to compute numerically the radial wave functions. Nevertheless they can be used, together with (3.2) for finding the derivative of the radial wave functions.

[FIGURE 3.3 OMITTED]

Programs. For the numerical simulations presented here we have used the commercial program MATLAB. The used source code can be obtained by request via e-mail to niurka@euler.us.esorran@us.es.

Acknowledgements. The authors thank J. Petronilho for interesting discussions as well as the unknown referee for his comments that help us to improve the paper. This research has been supported by the Programa de Acciones Integradas Hispano-Lusas HP2002-065 & E-6/03. The authors were also partially supported by the the DGES grant BFM 2003-06335C03-01 and PAI grant FQM-0262 (RAN); CMUC (JLC); and the DGES grant FIS2005-973 and PAI grant FQM-0207 (NRQ).

Appendix. The Laguerre polynomials. The Laguerre polynomials Ln' defined by the hypergeometric series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

These polynomials satisfy the following useful recurrence and differential-recurrence relations (see e.g. [1, 15, 17])

(A.1) d/dx [L.sup.a.sub.n](x) = [L.sub.a+1][n-x](x),

(A.2) x[L.sup.a+1.sub.n] (x) = (n + a + 1) [L.sup.a.sub.n] (x) - (n + 1) [L.sup.a.sub.n+1](x),

(A.3) x[L.sup.a+1.sub.n] (x) = (n + a) [L.sup.a.sub.n-1] (x) - (n - x) [L.sup.a.sub.n(x),

(A.4) [L.sup.a-1.sub.n] (x) = [L.sup.a.sub.n] (x) - [[L.sup.a.sub.n-1](x)

(A.5) (n + 1) [L.sub.a.sub.n+1](x) - (2n + a + 1 - x)[L.sup.a.sub.n] (x) + (n + a) [L.sup.a.sub.n-1](x) = 0.

* Received March 11, 2004. Accepted for publication June 3, 2005. Recommended by F. Marcellon.

REFERENCES

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[3] J. AVERY, Hyperspherical Harmonics: Applications in Quantum Theory, Kluwe Press, Dordrecht, 1988.

[4] V.G. BAGROV AND D.M. GITMAN, Exact Solutions of Relativistic Wave Equations, Kluwer, Dordrecht, 1990.

[5] R. BARRIO, B. MEEENDO, AND S. SERRANO, On the numerical evaluation of linear recurrences, J. Comput. Appl. Math., 150 (2003), pp. 71-86.

[6] P.K. BERA, S. BHATTACHARYYA, U. DAS, AND B. TALUKDAR, Recurrence relations for N-dimensional radial wave functions, Phys. Rev. A, 48 (1993), 4764-7.

[7] J. L. CARDOSO, Relacoes de Recorrencia para Funcoes do Tipo Hipergeometrico e Aplicacoes Mecanico-Quanticas, Master Thesis, Universidade de Coimbra, Coimbra, 1994.

[8] J. L. CARDOSO AND R. ALVAREZ-NODARSE, Recurrence relations for radial wave functions for the N-th dimensional oscillators and hydrogenlike atoms, J. Phys. A, 36 (2003), pp. 2055-2068.

[9] M. MARTIN NIETO, Hydrogen atom and relativistic pi-mesic atom in N-space dimensions, Am. J. Phys., 47 (1979), 1067-72.

[10] A. MARTINEZ-FINKELSHTEIN AND A. ZARZO, Varing orthogonality for a class of hypergeometric type polynomials, in Proc. 2nd. International Seminar on Approximation and Optimization, J. Guddat et al. eds., Approximation and Optimization Series, Peter lang, Berlin, 1995.

[11] A. MARTINEZ, A. ZARZO, AND R. YANEZ, Two approaches to the asymptotics of the zeros of a class of polynomials of hypergeometric type, (Russian), Mat. Sb., 185 (1994), pp. 65-78, translation in Russian Acad. Sci. Sb. Math., 83 (1995), pp. 483-494.

[12] Matlab, The Language of Technical Computing, the Math Works Inc Natick, MA, 2000.

[13] J. MORALES, J.J. PENA, AND J. LOPEZ-BONILLA, Algebraic approach to matrix elements: Recurrence relations and closed formulas for Hydrogenlike wave functions, Phys. Rev. A, 45 (1992), 4256-66.

[14] A. F. NIKIFOROV, S. K. SUSLOV, AND V. B. UVAROV, Classical Orthogonal Polynomials of a Discrete Variable. Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991.

[15] A. F. NIKIFOROV AND V. B. UVAROV, Special Functions of Mathematical Physics, Birkhduser, Basel, 1988.

[16] M. L. SANCHEZ, B. MORENO, AND A. LOPEZ PINEIRO, Matrix-element calculations for hydrogenlike atoms, Phys. Rev. A, 46 (1992), 6908-13.

[17] G. SZEG6, Orthogonal Polynomials, Amer. Math. Soc. Coll. Pub., 23, American Mathematical Society, Providence, Rhode Island, 1975.

[18] A. ZARZO, Ecuaciones Diferenciales de Tipo Hipergemetrico, Ph.D. Thesis, Universidad de Granada, 1995.

R. ALVAREZ-NODARSE ([dagger]), J. L. CARDOSO ([double-dagger]), AND N. R. QUINTERO ([section])

([dagger]) Departamento de Anolisis Matemotico, Facultad de Matemotica, Universidad de Sevilla. Apdo. Postal 1160, Sevilla, E-41080, Sevilla, Spain (ran@us.es).

([double-dagger]) Departamento de Matematica, Universidade de Tras-os-Montes e Alto Douro. Apartado 202, 5001 - 911 Vila Real, Portugal (j1uis@utad.pt).

([section]) Dpto. Fisica Aplicada I, EUP, Universidad de Sevilla, c/Virgen de Africa 7, 41011, Sevilla, Spain (niurka@us.es).

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Author: | Alvarez-Nodarse, R.; Cardoso, J.L.; Quintero, N.R. |
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Publication: | Electronic Transactions on Numerical Analysis |

Date: | Aug 1, 2006 |

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