_{1}

The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of operator disentanglement some operator identities are derived in an appendix. They allow calculating convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions.

Hermite and Laguerre polynomials play a great role in mathematics and in mathematical physics and can be found in many monographs of Special Functions, e.g., [

Laguerre 2D polynomials

written by application of an operator to the function

with the inversion (see also formulae (1.5))

Some special cases are

The differentiation of the Laguerre 2D polynomials provides again Laguerre 2D polynomials

and, furthermore, the Laguerre 2D polynomials satisfy the following recurrence relations

as was derived in [

The Laguerre 2D polynomials (1.2) are related to the generalized Laguerre (or Laguerre-Sonin)^{1} polynomials

that explains the given name. In most physical applications the second complex variable

The operators

and their powers can be disentangled (all multiplication operators stand in front of the differential operators) that using the explicit form of the Laguerre 2D polynomials (1.2) leads to the following operator identity

It is applicable to arbitrary functions

which is well known in quantum optics (transition from antinormal to normal ordering of boson creation and annihilation operators) and can be proved by complete induction but relation (1.9) can be also directly proved by complete induction.

The (special) Laguerre 2D polynomials

The Laguerre 2D polynomials are related to products of Hermite polynomials by [

and the inversion is

where the coefficients are essentially given by the Jacobi polynomials

and its inversion

and the application of the integral operator

and to

from which by comparison of the different representations follows

for Jacobi polynomials with equal upper indices specialized by argument

Clearly, Formulae (1.13)-(1.16) remain to be true if

Different methods of derivation of generating functions are presented in the monographs [

The operators which play a role in one of the definitions of Hermite and Laguerre (1D and 2D) polynomials are Gaussian convolutions and possess a relation to the Lie group

Hermite polynomials ^{2} (e.g., [

that leads to the following alternative definition from which results immediately the explicit representation

with the inversion

The definition (2.2) which is little known (see [

which can be applied to arbitrary functions

Such alternative definitions of a sequence of polynomials

with an arbitrary function

Generalized Laguerre polynomials

from which follows (compare with (1.2))

where

expansion of

Formulae (2.5) are closely related to the so-called umbral calculus [

Besides the well-known generating function for Hermite polynomials

in applications, in particular, in quantum optics of the harmonic oscillator the following bilinear generating function for products of two Hermite polynomials with equal indices but different arguments plays an important role (formula of Mehler; see, e.g. [

We represented here the right-hand side additionally in a sometimes useful factorization. This factorization is connected with the following identity (see [

and thus with a coordinate transformation. In the special case

it provides (see, e.g., [

which is a generating function for even Hermite polynomials and by differentiation with respect to variable x using

the corresponding generating functions for odd Hermite polynomials. We mention here that both generating functions (3.5) and (3.6) are not contained in the otherwise very comprehensive and impressive work [

Correct formulae of this kind one may find in [

In the limiting transition

As it is known this indicates a way for the introduction of Hermite functions

which are complete and orthonormalized according to

This underlines the great importance of the bilinear generating function (3.2). A proof of (3.2) can be given, for example, using an operational formula derived in Appendix A (Equation (A.12) there) or using (3.3) in connection with the generating function for generalized Laguerre polynomials that shifts it to the proof of (3.3) (the more direct proof of (3.2) in [

In the derivation of generating functions for Laguerre 2D polynomials

for arbitrary complex

guerre 2D polynomials up to factors

The following generating function for Laguerre 2D polynomials is easily to obtain from the alternative definition of

in particular for

As an intermediate step to the generating (4.2) one may consider the generating function with summation over only one of the indices in the special Laguerre 2D polynomials

which is to obtain analogously to (4.2) using the first definition of these polynomials in (1.1). Written by the usual generalized Laguerre polynomials according to (1.7) this provides

which after substitutions

We now calculate the basic generating function for the product of two special Laguerre 2D polynomials. Using the first of the definitions in (1.1) which corresponds to the alternative definition of the Hermite polynomials in first line of (2.2), we quickly proceed as follows (remind (4.1))

where in third step it is used that

This result for the generating function can be factorized in the following form

in analogy to the formula of Mehler (3.2) and a fully analogous derivation of this formula by coordinate transformations is possible. In Section 10 we derive such a decomposition of the product of two Laguerre 2D polynomials with the same indices but different arguments which provides a further insight into the factorization in

(5.2) according to (5.3). If we substitute in (5.2)

then we find by this limiting procedure from (5.2) the generating function (4.2).

Expressed by usual generalized Laguerre polynomials, relation (5.2) takes on the following forms

where on the left-hand side relations (1.7) were used.

Similar to the case of Hermite polynomials, we now make the limiting transition

the variables. With

where

which are complete and orthonormalized according to

Herein,

whole complex plane. Relations (5.7) and (5.8) can be used for the expansion of functions of two variables in Laguerre 2D polynomials or Laguerre 2D functions.

By forming derivatives of (5.2) with respect to the variables, one can derive related formulae. Furthermore, by specialization of the variables, we obtain new formulae. For example, due to

This is the well-known generating function for the usual Laguerre polynomials

that is the convolution of two 2D Gaussian functions (see Appendix A, Formulae (A.22) or (A.26) with corresponding substitutions) which provides the result on the right-hand side of (5.9).

Special Hermite 2D polynomials

The generating function for products of two special Hermite 2D polynomials defined by (6.1) factorizes

and is easily to obtain by using explicitly the Mehler Formula (3.2) with the result

In special case

that is the product of two generating functions for even Hermite polynomials of the form (3.5). By differentiations of this formula with respect to variables x or y one finds also the cases for odd Hermite polynomials and the mixed case of even and odd Hermite polynomials but due to factorization this belongs already to the primary stage of the generating functions (3.5) and (3.6).

Sometimes in quantum-optical calculations, one has to evaluate the following double sum of the form of a generating function for products of special Laguerre 2D with special Hermite 2D polynomials the last defined by (6.1) and can quickly proceed to the following stage

Here we have two possibilities to continue the calculations. Using two times the generating function for Hermite polynomials, we obtain from (7.1)

and using first the generating function for Laguerre 2D polynomials, we find

The remaining problem is to calculate the two-dimensional convolutions in (7.2) or (7.3). Both convolutions can be accomplished using auxiliary formulae prepared in Appendix A. The result is

The complexity of this generating function finds a simple explanation to which we say some words at the end of this Section.

In the special case

where, in addition, a factorization is given. Using the generating function (3.5) for even Hermite polynomials we get from this factorization the identity

which we consider from another point of view in Section 9 providing thus some better understanding for it.

For

where we used the identity (A.15) in the Appendix A in special case

which with obvious substitutions (

We mention yet that expressed by the usual generalized Laguerre polynomials according to (1.7) and by applying the doubling formula for the argument of the Gamma function the left-hand side of (7.5) can be written

the last by symmetry of the Laguerre 2D polynomials or using (1.7).

As already mentioned in the Introduction the special Laguerre 2D and Hermite 2D polynomials can be combined in one whole object of polynomials

We now consider a set of simple (in the sense of not double!) sums over products of Laguerre 2D polynomials with two free indices

The last two steps of making the argument displacements and using the scaling property (4.1) lead to the following final representations (among other possible ones)

Expressed by generalized Laguerre polynomials using (1.7), this relation takes on the form (compare a similar form in [

In the special case

or expressed by the generalized Laguerre polynomials after division of (8.3) by

where we made the substitution

In most representations of orthogonal polynomials, one can only find generating functions for products of generalized Laguerre polynomials where the upper indices are parameters and are not involved in the summations (e.g., [

and substituting

where

The sums over products of Laguerre 2D polynomials (8.2) or (8.3) possess proper importance for sum evaluations which sometimes arise when working with these polynomials. They also form a partial result on the way to the evaluation of the generating function (5.2) for the product of two Laguerre 2D polynomials. Taking in (8.2)

the special case

Joining herein the two exponential functions we see that the right-hand side of (8.8) is equal to the right-hand side of (5.2) as it is necessary and thus we have calculated here this generating function in a second way.

We illuminate now a cause for the possible factorization in the generating function (7.5) for Laguerre 2D polynomials with even indices. For this purpose we make in (7.5) the substitutions

Therefore

that can be written

where the alternative definition of Hermite polynomials in (2.1) is applied and where an apparently unknown sequence of finite sum evaluations

is inserted. These sum identities are proved already by the obvious equivalence (7.5) and are easily to check for small

We derive here a decomposition of the generating function for an entangled product of two equal Laguerre 2D polynomials with different arguments into a product of two simple generating function for Laguerre 2D polynomials which in its further application leads immediately to the given factorization in the generating function (5.2) and provides a certain explanation for it. The considerations are in some sense similar to the considerations in the previous Section.

We now make in (5.2) the following substitutions of variables

with the inversion

from which follows for the operators of differentiation

As a consequence we find in transformed coordinates (see also (5.1)) with application of formula (A.26) in the Appendix A for the evaluation of the Gaussian integrals

that using the generating function (5.9) for simple Laguerre 2D polynomials with equal indices can be written

If we go back on the right-hand side to the primary variables

It is not possible to say generally which of the two alternative definitions of Hermite polynomials in Section 2 and of Laguerre 2D polynomials in Section 1 are better to work with. This depends on the problem and sometimes also on the taste and the former experience of the user. We demonstrate this in the simplest case of the derivation of the generating function (3.1) for Hermite polynomials. Using definition in the first line of (2.1) we calculate

and using definition in the second line of (2.1)

In the first case we use that

The alternative definitions of Hermite (1D and 2D) and Laguerre 2D polynomials extends the arsenal of possible approaches to problems of their application and one should have for disposal the new method in the same way as the former methods.

We have derived and discussed generating functions for the product of two special Laguerre 2D or Hermite 2D polynomials and for the mixed case of such products. In our derivations, we preferred the first (operational) definition of the Laguerre 2D polynomials (2) from the two alternative ones given in (1.2) which as it seems to us is

advantageous for this purpose. This is due to the separation of the same operator

tions for the monomials

In Section 11 we demonstrated the differences between both methods in one of the most simple cases which is the derivation of the well-known generating function (3.1) for Hermite polynomials. However, our main aim was the derivation of new bilinear generating functions. In the bilinear generating functions for Hermite polynomials (3.2) as well as for Laguerre 2D polynomials (5.3) we found interesting factorizations which establish connections to more special (linear) generating functions for these polynomials with transformed variables. Due to the rudimentary character of the generalized Laguerre (-Sonin) polynomials

From the bilinear generating functions or generating functions for products of Laguerre 2D polynomials one can derive completeness relations for the corresponding polynomials in the way such as demonstrated. Furthermore, we derived a simple sum (in the sense of not double sum!) over products of special Laguerre 2D polynomials which can be taken as intermediate step for the derivation of the generating function but this formula possesses proper importance for other calculations and was already useful in an application in quantum optics of phase states. The number of generating functions and of relations for Laguerre 2D and Hermite 2D polynomials is relatively large and a main source for suggestion are known generating functions and relations for usual Laguerre and, in particular, for Hermite polynomials.

The three generating functions for products of Hermite 2D and Laguerre 2D polynomials (5.2), (6.3) and (7.4) can be considered as special cases of generating functions for products of Hermite 2D polynomials

We hope that we could convince the reader of some advantages of the use of Laguerre 2D polynomials in comparison to usual generalized Laguerre(-Sonin) polynomials as their radial rudiments. Although the usual Laguerre (and also Hermite) polynomials are mostly present in readily programmed form in mathematical computer programs it is not difficult to programme in the same way the Laguerre 2D polynomials by their explicit Formulae (1.2) as finite simple sums.

A main region of application of the derived generating functions is quantum optics of two harmonic oscillator modes and of quasiprobabilities of oscillator states such as the Wigner function and in classical optics the theory of Gauss-Hermite and Gauss-Laguerre beams and we applied some of the here derived relations in papers of former time [

In the following Appendix A, we develop some basic operator identities which are useful for calculation of convolutions of one- and two-dimensional Gaussian functions in combination with polynomials and which were used in most of our derivations of the generating functions. The corresponding operators are connected with the Lie group

AlfredWünsche, (2015) Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials. Applied Mathematics,06,2142-2168. doi: 10.4236/am.2015.612188

We sketch in this Appendix the derivation of some mostly novel and useful operational formulae related to convolutions of Gaussian functions of one and two variables and use for this purpose the technique of operator disentanglement of

As canonical basis of the abstract Lie algebra to

As first case, we consider the following realization of the operators

Using the commutation (r is scalar parameter)

we obtain the following relation (s is a second scalar parameter)

Now, we can apply the following disentanglement relation for general group elements of

where

realization or

bra to the inhomogeneous symplectic Lie group

With the specialization

and the disentanglement searched for is

Inserting this into (A.4) and going back to realization (A.2), we obtain the following important operator identity (we use

If we apply this operator identity to an arbitrary function

from which follows that ^{3}

By applying the operator identity (A.9) to an arbitrary function

with substitution of variable

which is a sometimes useful transformation of (A.12).

Choosing the function

which as shown may also be written as the convolution of two normalized Gaussian functions and provides as result again a normalized Gaussian function as it is well known (notation “*” means forming the convolution of two functions). The operator

where due to

but if we shorten the fractions then the correlation of the signs of the now two different roots becomes unclear^{4}. In application to the Hermite polynomials

where again one can choose an arbitrary sign of the roots

As second case, we now consider the following realization of the operators

Our notation of the two independent (in general, complex) variables

follows

The operator which we have to disentangle corresponds to the special choice

Using the disentanglement relation (A.5), from (A.19) follows the important operational identity

Applied to arbitrary functions

which in special case

as some useful transformation of (A.22).

We mention that the monomials

from which follows in application of its exponential to an arbitrary function

in analogy to formula (A.11) together with (A.10). This was used in (A.22).

In special case of function

which provides again a normalized Gaussian function with the parameter

The full power of (A.22) is seen if we apply it to the functions

Applied to Laguerre 2D polynomials

In last two relations one has to choose an arbitrary but the same sign within one formula for the roots

We mention that an additional displacement of the arguments does not make any difficulties in the derived operator relations. For example, according to

we can generalize the operator identity (A.9) in the following way

and in application to an arbitrary function

Choosing

one finds the transformation

and by multiplication of both sides with an arbitrary function

that are two different representations of the convolution of a normalized Gaussian function with an arbitrary function

Analogously to (A.29), relation (A.21) can be generalized including two, in general, different complex displacements

with the consequence

By a limiting procedure in analogy to the derivation of (A.34) one obtains from (A.36) for

which are two different representations of the two-dimensional convolution of a normalized Gaussian function with an arbitrary function, in particular

with the two-dimensional delta function obtained by the limiting procedure

in analogy to (A.32). All this can be also obtained by Fourier transformation and its inversion.

The displacement operation (A.29) with substitutions

Necessarily, this agrees with the specialized generating function (7.5). We used this formula for the evaluation of (7.2) in one of some possible variants of proof. In application of the operator

where we applied the Taylor series of

which for

The derived identities are particularly important in connection with the operational definition of the Hermite polynomials in (2.2) and of the Laguerre 2D polynomials in (1.2). It seems to us that the here chosen derivations belong to the most simple and perspective ones.