Let us suppose that we know the Taylor series of a function
It is not rare that we want to evaluate a new primarily unknown function given by its power series of the following kind
with coefficients taken from the Taylor series of in (1.1). This can be made by a linear integral transformation of the function which is closely related to a Laplace transformation and which is now also called Sumudu transformation1. It was introduced by Watugala  and further developed by Belgacem and coauthors, e.g.,      and many others, e.g.,  . The article  contains basic examples of Sumudu transforms of functions. Since Laplace transformations are widely applied in mathematics for the solution of ordinary differential equations which have important applications in physics (e.g., electrical engineering) and in other sciences and since there exist detailed investigations of these transformations and voluminous tables, e.g.,      it seems to be possible to enlarge the store of power series which can be evaluated in closed form. The present article applies the Sumudu transformation mainly to obtain new Generating functions for Hermite polynomials. In the following we derive and explain this integral transformation and consider in Section 5 a modified similar transformation which may be considered in a loose sense as half the part of the way from the original function to its Sumudu transform. In Section 3 we demonstrate how Sumudu transformation can also be considered as a limiting case of Fractional integration.
Using the well-known sequence of integrals with the real parameter
we obtain from the Taylor series of the following general identity
and, finally, if we substitute herein (1.1)
1Sumudu means “smooth, noble” in Sanscrit and in Sinhala, the language of Sinhalese people of Sri Lanka.
We inserted in (1.4) according to (1.3) and changed the order of summation and integration due to supposed absolute convergence of the sum. From (1.5) it can be seen that the functions have to be determined only for . Setting with the Heaviside step function ( for , for ) in the sense of generalized functions one may extend the integration in (1.5) from to that makes some proofs easier, e.g., inversion of the transformation.
By the substitution of the parameter we obtain from (1.5) the form
If we denote the standard form of the Laplace transformation by  
the transition is found to be a modified Laplace transformation in the following sense
where stands here for Sumudu transformation. To each pair of original function and its Laplace transform which one may calculate or can find in tables (e.g.,   ) one obtains a pair of Taylor series of and if one of these series is known2.
By the substitution with x as positive parameter and using partial integration we can bring (1.5) to the form
which by partial integration leads to
that means again to the starting form (1.5). We wrote here the multiplication of the argument of a function by a factor in operator form as  3
2The last cited tables  contain the so-called Laplace-Carson transforms and thus they are related with the standard Laplace transforms simply by .
3We mention that the operators with the commutation relations form a basis of a realization of the Lie algebra to the Lie group with useful applications to Hermite polynomials.
On the other side we may consider the function in (1.9) as multiplication of the argument x of the function by a factor u and using then again (1.11) we can represent (1.9) in the form
The operator which transforms into and which has to be applied to is here fully separated from and represents the Mellin transformation  (chap. VI) of the function with argument substituted by
the operator . The Mellin transform of a function is defined by
and its special case of the function leads to
and may be considered as one of the possible basic definitions of the Gamma function. Extensive tables of Mellin transforms of functions we find in cited work of Bateman and Erdélyi  . Beside more insight into the nature of Sumudu transformations we hope that the representation (1.12) can be applied in future also more directly using the many known formulae for the Gamma function.
We now consider the inversion of the transformation
As it is known the inversion of the Laplace transformation (1.7) is (e.g.,  )
where real c is widely arbitrary but must be chosen in the region where the integral is convergent. For the Sumudu transformation (1.5) written in the form (1.6) as a Laplace transformation of this leads to the inversion
This transformation allows to evaluate the Taylor series of a function from the known function if the integral (1.17) can be evaluated.
As a first example which gives a deeper insight into the nature of the Sumudu transformation we consider the functions . Using (1.5) together with (1.3) one calculates that these functions are transformed into functions according to
that due to linearity of the transformation can be immediately applied to the Sumudu transformation of Taylor series. This correspondence possesses yet another aspect. For this purpose we mention that is eigenfunction of the
operator to eigenvalue and we conclude
and from this (existence of Taylor series assumed)
The second part of (1.19) follows immediately from the representation (1.12) of the Sumudu transformation. However, this representation (1.12) alone is not very useful for summing up Taylor series as result of the transformation if we do not possess in addition integral representations of the form (1.5) which allow, at least in principle, to sum up the arising series by calculating an integral.
As generalization one may consider a transformation of the kind
where is an arbitrary given function and the function to transform.
Since are eigenfunctions of the operator to eigenvalue n from (1.20) follows for
For the evaluation of arising new Taylor series after the transformation it is necessary to possess in addition to (1.21) an integral representation (or an equivalent of it) which allows to calculate more directly from given . In particular, this is also true for the form (1.12) of the Sumudu transformation. In this case as mentioned we have it in the basic form (1.5) of the Sumudu transformation.
2. Some General Rules for Sumudu Transformation
In this Section we derive and discuss some general rules for Sumudu transformations according to (1.1) and (1.2) with their explicit form given in (1.5) and denote the correspondence between and by
Then the correspondence of a function with stretching parameter a of the argument to its Sumudu transform according to
is determined by
where denotes the Sumudu transform of according to (2.1). Thus the multiplication of the argument of a function by a number a leads to a simple relation of its Sumudu transform to the Sumudu transform of the primary function and it is not necessary to consider separately Sumudu transformations of functions . For the case of an argument translation of a function this is a little more complicated (see below).
To the differentiation of a function corresponds a function using partial integration according to
with the result of this correspondence
if the upper boundary does not give a contribution. In
analogous way one finds after a few calculations for low numbers of n the general formula ( )
It is not difficult to prove it by complete induction.
For the displacement of the argument of a function by as a parameter
we find with intermediate substitution of the integration variable for the correspondence to the Sumudu transform
The second sum term is some kind of incomplete Sumudu transformation of the function and it vanishes for . By the substitution this can be transformed to
We mention that for even functions with existing Taylor series we find the correspondence
and for odd functions the correspondence
Sumudu transformation preserves the lateral symmetry of a function.
3. Fractional Integration and Sumudu Transformation as a Certain Limiting Case
We now consider fractional integration and differentiation (e.g., Bateman and Erdélyi , Vladimirov , Oldham and Spanier , Samko, Kilbas, Marichev ) which is often useful in connection with Sumudu transformation. Furthermore, Sumudu transformation can be represented as a certain limiting case of fractional integration as we will show in this Section. First, we shortly introduce fractional integration.
It is known (in an elementary form, e.g., Kranzer , p. 205) that the n-fold integration of a function can be represented by a convolution integral as follows
This can be proved by complete induction first changing the order of integration in the arising double integral and then calculating the inner integral
that together with the correct as initial member proves the proposition.
The n-fold integral (3.1) can be generalized in the rank of a definition to a ν-fold fractional integral with arbitrary real parameter by the extension in (3.1) (positive real solution is to insert for the multi-valued function )
This form is also called the Riemann-Liouville integral  . The case as limiting case provides
The convolution (3.3) of with powers of x is one of the possible basic definitions of fractional integration of order of a function , e.g.,     (the last two  and  with a short historical introduction). In  (chap. XIII) and in  (pp. 140-142) one finds tables of fractional integrals. Fractional derivatives ( ) can be obtained from appropriate fractional integrals ( ) by corresponding integer differentiations. Interesting is the view from the theory of the convolution algebra of generalized functions where one can introduce the functions which generalize the step function  (p. 142; he uses instead of our ) by
and where “ ” stands for the convolution . Fractional integration of functions can be then determined by
with the consequence
that means associativity as it has to be for successive convolutions using a notation of the form of multiplications.
Sumudu transformation and fractional integration of order do not commute. If we make first the fractional integration of order of the powers with symbol and then with the result the Sumudu transformation we obtain
and if we make first a Sumudu transformation of the powers and then a fractional integration of order of the result we find
In comparison, two successive fractional integrations of order and are commuting and result in a fractional integration of order and, therefore, form a one-parameter group with the identity operation belonging to .
By comparison of the Laplace transforms in the tables of Bateman and Erdélyi  (pp. 120-204, here, in particular, p. 121, (8) and (9)) with the Sumudu transforms (2.6) one finds that essentially the variable of the Laplace transform is substituted by its inverse in the Sumudu transform. In case of negative values
we have in front of the powers which are singular
functions of x at where the singularities in the sense of generalized functions have to be determined more precisely.
We now consider a certain limiting transition from fractional integration to Sumudu transformation and write for this purpose (3.2) first in the way
and substitute with that leads to the following relation
If we now make the limiting transition using the definition of the exponential functions by the limiting process
and known already to Euler we obtain from the right-hand side of (3.11) and in agreement with (1.5)
From the left-hand side of (3.11) in the limiting transition using the well-known Stirling approximation of the Gamma function for
inserting this for we find
Thus we may write the Sumudu transformation as the following limit of the fractional integration if we substitute in addition
The right-hand side of this relation is one of the ways to represent the Sumudu transformation by a limiting procedure from fractional integration.
4. Examples for Sumudu Transformations
We provide now a few simple examples for Sumudu transformations. The transformation of the “basic” monomial functions was already considered in (1.18) and (1.19) and leads to . It illustrates the action of
the operator in (1.12) in ideal way.
The most striking genuine example for a Sumudu transformation establishes the connection between the Exponential and the Geometric series, clearly, with a well-known result. One finds 
For this example inserting on the right-hand side of the formula
(1.17) it is easy to demonstrate the inverse Sumudu transformation from the Geometric to the Exponential series according to
Here we applied the residue theorem for the Cauchy integral over a closed contour C in the complex s-plane including the only singularity of the integrand at . The contour C consists for parameter of the line through parallel to the imaginary axis closed by a semicircle with
or (i.e. to the left of the line) and followed by
where the integral over the semicircle then vanishes. The possible real values c in the integral are then only restricted by for the purpose to include the singularity.
A further interesting simple example establishes a connection between the Bessel function and the Exponential function in the following way 
In this example it is merely the integral transform from the Bessel function to the exponential function which is less known (but not unknown) in comparison to the other results. Clearly, by substitution one may obtain similar formulae for the Bessel function .
Related to (4.1) one finds
and related to (4.3)
For integer one finds from this relation
It is here again the integral over modified Bessel functions multiplied with an exponential function which is less known than the involved Taylor series. We mention that the n-th derivative of the Bessel function leads to the
entire function according to the more general formulae
This can be generalized by fractional differentiation from integer n to arbitrary positive real . The corresponding more general formulae for fractional integration of order are  (chap. 13.1. formula (63) in the table) or  (p. 142, Eq. 9)
The integrals (1.3) cannot be extended from positive to negative n since . Nevertheless, the Sumudu transform can be calculated for some functions without a Taylor series at and it is interesting to see in an example what happens. For this purpose we calculate the Sumudu transform of the function where is the Heaviside step function . We find
The Sumudu transform is singular at with the Laurent series written in (4.9). If we formally write the inverse Sumudu transformation to
adding for negative powers of x in the denominator the factor
then for we have and the series (4.10) becomes undetermined in the neighborhood of due to the singularities.
The Sumudu transformation is an interesting addition to other methods of the analysis of functions but often one possesses alternative possibilities to evaluate infinite series.
5. A modified Sumudu or a Kind of Gauss Transformation
In analogy to the integral (1.3) we now consider the sequence of integrals for non-negative integers with real positive parameter x (identity )
which after simple substitutions is related to (1.3) and extends it practically to semi-integers n. We used in (5.1) the well-known formula for doubling of the argument of the Gamma function
with . Two special cases of the sequence of integrals (5.1), first, setting leads to
with for real and, second, setting leads to
The integral (5.1) can be used to transform a function with the Taylor series
in analogy to (1.5) into a function in the rank of its definition by the Taylor series as follows
This transformation can be compactly written in the integral form
This is a certain kind of a Gauss transformation of the function 4. If we are able to find the integral on the right-hand side for a function we may
evaluate the infinite power series denoted by on the left-hand side.
In the same way as in case of the Sumudu transformation we now substitute with as a parameter and find from (5.7)
4As Gauss transformation is sometimes denoted a transformation with displacement of the argument of the Gauss function according to  (chap. 10.13 (30)). In other sources  (from p. 170 on) almost this same transformation with the kernel from the heat equation with choice is called a Weierstrass transform.
and apply the operator representation (1.11) to the function with u as a factor to the variable x of the function . In this way follows from (5.7)
In analogy to (1.12) this can be represented by a Mellin transformation of the function taking into account
with the argument s substituted by the operator . This explains the upper index of our notation for the transform of .
If we substitute the integration variable in (5.7) then the function with variable substitution takes on a form which can be written
The right-hand side of this relation possesses the form (1.5) of a Sumudu
transformation of the function with leading to a new function involving the function and determined by the left-hand side of
(5.11). The inversion of the transformation according to (1.17) with corresponding substitutions is
or with variable substitution back to the original function variable
We now give a few general rules for the considered transformation. The multiplication of the argument of a function by a number a leads here as in the case of Sumudu transformation (2.2) again to
where is the corresponding Gauss transformation (5.7) of the function .
For even functions or combinations of them from (5.3) follows
and for the odd case using (5.3)
The considered Gauss transformation preserves the lateral symmetry of a function.
6. Examples for Modified Sumudu or Gauss Transformations
As an example for the integral transformation (5.7) we consider the case
which belongs to the Exponential function as the primary function. For the evaluation of the corresponding series we find then
The integrals on the right-hand side are well-known Special functions which can be represented by the Error function in the following way (we
where the Error function “ ” is defined by5
and is the Complementary error function and, furthermore, the Confluent Hypergeometric function. The Taylor series of both of the symmetric and of the antisymmetric parts of are well known.
In the next two examples we use the Generating function for Hermite polynomials
as starting point for our modified Sumudu or Gauss transformations according to (5.15) and (5.16). Their special cases for even and odd Hermite polynomials are immediately to obtain from (6.5) by separating the symmetrical and the antisymmetrical part with respect to variable z
5In older sources (e.g., , chap. 9.9.) the Error function is defined under the same notation without the factor but in the modern definition (6.4) with the consequence it is programmed in “Mathematica” (now mostly denoted “erf(z)”).
where the variable z is a parameter. From (5.15) we find then
and from (5.16)
The integrals in (6.7) and in (6.8) can be evaluated (see Appendix A). From (A2) follows with corresponding substitutions the simple result
and from (A3)
With the substitution one obtains from (6.9)
and from (6.10) with definition of the Error function of imaginary argument
The generating function (6.11) can also be obtained from the following generating function for the product of two Hermite polynomials (formula of Mehler, , chap. 10.13., Eq. (22); see also )
by setting and using
It seems that the generating function (6.10) or equivalently (6.12) is new and is not easily to obtain by other methods (e.g., integration of (6.13) with respect to variable y and limiting transition).
We mention that the Sumudu transformation of the basic generating function (6.5) for Hermite polynomials according to (1.2) to a function
with parameter z leads to a divergent series which turns out to be an asymptotic series of a function which can be evaluated. We make a few remarks about this in next Section. However, one may pose the problem of a transformation also in
opposite direction that means to evaluate the series using (6.5)
that is equivalent to an inverse Sumudu transformation. This we deal with in the Section after the next.
7. Sumudu Transformation of Hermite Polynomials and of Their Basic Generating Function
Since Sumudu transformations transforms powers into the Sumudu transformation of Hermite polynomials transforms them according to their explicit representation as follows
Such transformations may easily be written down in similar way for all other explicitly given polynomials. As a rule, such truncated series are not introduced as Special functions with a special function symbol and, clearly, problems of their convergence also do not exist. In contrast, the case of infinite series, in particular, the Sumudu transformation of generating functions can make problems.
We now consider the Sumudu transformation of the basic Generating function for Hermite polynomials
with respect to the variable t and with z as a (in general, complex) parameter. In Appendix B we develop an operational technique to work with the operator
in combination with exponential functions. If we use the
representation (1.12) of the Sumudu transformation of the Generating function of Hermite polynomials with respect to variable t that means to
we have to look for the interpretation of the following function ,
whereas the commutativity of the two integral operators and is fully obvious the change of the order of summation and of the
application of the two integral operators is afflicted with some problems because the arising series are no more absolutely convergent. We assume in the following that changing of these operations is possible and that the arising conditionally convergent series can be interpreted after the evaluation as some generalized functions but do not discuss the last in detail. If we apply the Taylor series of the
operator to the function we obtain
It is obvious that this sum is divergent for arbitrary since from a certain m on the factor grows faster than which last is the quotient of two neighbored terms and . On the other
side if we directly make the Sumudu transformation of the Generating function using (1.9) we find
This can be considered as the Sumudu transform of the basic Generating function of Hermite polynomials and (7.5) is its asymptotic expansion.
Since the considered new Generating function for Hermite polynomials as mentioned possesses problems of convergence one may also view onto these functions as generalized functions with weak convergence that means as linear continuous functionals over an appropriate space of basis functions. The Geometric
series (in our case ) is not convergent for
and its sum is singular at . Within the theory of generalized functions one has to determine this singularity more precisely, for example, as principal value
or as and in correspondence with this their transforms. We do not try to discuss this more exactly.
8. A New Kind of Generating Function for Hermite Polynomials by Inversion of the Sumudu Transformation
In last Section we saw that the Sumudu transformation of the basic Generating function for Hermite polynomials leads to the series
which is not convergent for arbitrary . We also saw that this must not necessarily mean that we cannot give it a sense, for example, as some generalized
function since it is formally equal to applying the integral operator to the geometric series changing the order of summation and application of the integral operator.
Let us now assume that we want to evaluate the following series
where are the Hermite polynomials and z is a parameter and where t is the variable involved in the transformations. This is equivalent to an inverse Sumudu transformation of the basic generating function (6.5) for Hermite polynomials which we write now
if we consider as Sumudu transform of in (8.1). However, we try to calculate in (8.1) directly from its definition. We find here some relations between Hermite polynomials and series of them to categories of Bessel functions with integer indices and also to Laguerre polynomials.
If we use the following alternative definition of Hermite polynomials (e.g., )
which generates the Hermite polynomials by an integral transformation of the power functions and can be represented as a convolution (integration path in w-plane widely deformable) we quickly arrive at
where is a standard Bessel function. If we use the Taylor series of the integral operator we may transform (8.4) according to
A representation in operator form which is equivalent to the right-hand side of (8.5) can be directly obtained using the Rodrigues definition of Hermite polynomials as follows
where the last line means the application of an operator function . We mention in this connection also the following disentanglement formula for the
operators which is an operator identity
Applied to it leads back from the right-hand side of (8.6) to its left-hand side.
In the special case one finds from (8.1)
Apart from a special case of the rarely used Hypergeometric Function the right-hand side does not belong to the Special Functions for which an evaluation in closed form is introduced and investigated in literature. This means that one cannot expect a closed representation for the more general series on the right-hand sides in (8.4) or (8.5). Using the formula (1.17) for the inverse Sumudu transformation and the generating function (8.2) for Hermite polynomials one obtains the following integral representation of the function in (8.1)
The closed evaluation of the integral by a known function is apparently unknown and, as discussed, can hardly be expected but, nevertheless, the transformed series are important because their convergence properties are different from that for the original series (8.1) and may lead to different approximations.
We will make now some further consideration to the series on both sides of (8.5). If we start from this function then the basic generating function for Hermite polynomials (8.2) is its Sumudu transform and one may expect that we can derive some identities. According to the Sumudu transformation (1.5) we have here specialized
If we change on the left-hand side the order of integration and summation we find that we do not have a correspondence alone of each sum term over m on the left-hand side and over n on the right-hand side. Making the substitution of the integration variable on the left-hand side we obtain
This transformation separates the underbraced integrals which depend only on the variable combination 2zt. However that the integrals
are also independent on m is not easily seen. Supposing the knowledge of the independently derived identity (8.5) and of the generating function (8.11) for Hermite polynomials one may even conclude onto the given evaluation of the underbraced integral.
Starting from (8.5) one may derive closed representation of the series by an integral which, however, also does not lead to a full evaluation by known functions.
We write (8.5) in the following form with and apply formula (5.3)
If we apply here the Taylor series of an analytic function and its symmetrization
We may represent (8.13) in the following integral form
Apparently, an evaluation of the integrals on the right-hand side by well-introduced Special functions is unknown.
9. Representation of the New Generating Function for Hermite Polynomials by Series of Laguerre Polynomials
We derive now another representation of the function (8.1) and start from the Rodrigues definition of the Hermite polynomials within the series, an approach which becomes useful for large values of and thus for the asymptotic case
By interchanging the order of summations in the double sum we find
where denotes the Laguerre polynomials, explicitly
It is well known that the Laguerre polynomials are a special case of the Confluent Hypergeometric function defined by
according to (e.g., , chap. 10.12., Eq. (14))
where, clearly, is not a polynomial but an entire function represented by the series
The transformation of the Confluent Hypergeometric function
used in (9.5) in specialized form is known as the Kummer transformation ( , chap. 6.3.). The Kummer transformation possesses involutory character that means applied to the right-hand side provides anew the left-hand side.
If one uses (9.5) one may represent (9.2) also in the form
Here and equivalently in (9.2) we separated the factor in the evaluation of the left-hand sides and obtained in this way essentially different representations of the considered kind of generating functions. However, again we are not able to evaluate this generating function by known Special functions and transformed it only into another form of series with the perspective to make other approximations than before.
10. Two Further Kinds of Generating Functions for Hermite Polynomials
In comparison to (8.1) it is interesting to consider two slightly different Generating functions (or series) for Hermite polynomials as follows, first
Using the alternative definition of Hermite polynomials (8.3) one may transform (10.1) according to
and (10.2) according to
In the special case with the special values (6.14) of the Hermite polynomials in this case and using the Hypergeometric function one finds from (10.1)
and from (10.2)
This suggests that one cannot write the evaluation of the general series (10.1) and (10.2) in closed form by well-introduced functions.
Inserting the Rodrigues definition of the Hermite polynomials in (10.1) we obtain by a calculation in full analogy to (9.1) and (9.2)
and inserting it in (10.2)
The relation of the Associated (or Laguerre-Sonin) polynomials to the Hypergeometric function together with the Kummer transformation (9.7) of is
The primary summation is here substituted by another summation with excerption of the factor but a full evaluation of the series by well-established functions was not obtained.
Since a full evaluation of the series for the considered Generating functions could not be found we will make a transition to an integral representation in analogy to (8.15). To shorten slightly the notation we make the same substitution
as in last Section. Then we find by Taylor series expansion of the operator from (10.3) at first
and analogously from (10.4)
Then using (5.3) and (8.14) we obtain the following possible representations from (10.10)
and from (10.11)
The representation by Bessel functions of semi-integer index can be taken from (10.3) and (10.4) by comparison with (10.12) and (10.13) and it is interesting to compare it with formula (8.15). The different representations of the Generating functions by series and integrals may be taken as starting point for different approximations.
It was shown that Sumudu transformation of a function as a relative of Laplace transformation can lead to the evaluation of series by integral transformation of known series. In Section 5 we considered a modified Sumudu transformation of which contains the Gaussian bell function instead of the Exponential function and can also lead to the evaluation of series by transformation of known series. The most striking example of a Sumudu transformation establishes a connection between the Exponential and the Geometric series. The use of Sumudu transformation in connection with fractional integration is often useful. This extends the arsenal of methods for the evaluation of infinite series or sometimes of integrals apart from other possible applications, for example, in the theory of differential equations. Furthermore, we derived a certain limiting transition from fractional integration to Sumudu transformation.
For the illustration of the Sumudu transformation we gave simple basic examples and applied it in Sections 7 - 10 to obtain new Generating functions for Hermite polynomials. To each function with known Laplace transformation and known Taylor series one may construct a new series which one may evaluate. It is difficult to predict whether or not such a programme will be realized for the great number of Laplace transformations of functions which one may find in the published tables (e.g.,   ). In Section 6 we examined examples for the considered Gauss transformation leading to new Generating functions for Hermite polynomials. In Section 7 we investigated a Sumudu transformation to a new kind of Generating functions for Hermite polynomials which leads to an asymptotic series. In Sections 8 and 9 we considered a series for a new Generating function for Hermite polynomials which corresponds to an inverse Sumudu transformation of the basic Generating function for Hermite polynomials. However, this leads only to transformations of series without the full evaluation of these series by known Special functions. We hope that these examples will find also applications in physics and other sciences.
Almost a decade ago we independently found a transformation which figured in our records as modified Laplace transformation. However, a few years ago we realized that it exists already under the name Sumudu transformation. Since it is closely related to Laplace transformation and the basic integral (1.3) is well known, likely, it was independently discovered several times.
Evaluation of Two Related Gaussian Integrals
For convenience although, in principle, known we calculate here two integrals used in Section 6.
With two parameters a and u we have
From this follows in the upper case
and in the lower case
The calculation of the upper and lower case of the integrals in (A1) have to lead to a symmetric and an antisymmetric function with respect to variable u, respectively.
Application of Two Known General Operator Identities to Hermite Polynomials and Its Basic Generating Function
For application to the calculation of the basic Generating functions for Hermite polynomials we collect in this Appendix two known operator identities without their proofs which can be found in monographs about group theory and its representation theory and specialize them.
For two arbitrary linear operators A and B in a (in general, infinite-dimensional) linear space the following operator expansion is true (e.g., Louisell , p. 136)
with denoting the commutator of A and B.
The second operator identity which we consider here is a special case of the general product of two exponential operators which is dealt with in group theory in the transition from a Lie algebra to its Lie group by exponentiation of the general Lie algebra operators and is the content of the Baker-Campbell-Hausdorff-Dynkin formula (e.g., Kirillov , p. 119). The here considered special case is (e.g., Messiah , Glauber  and Louisell )
It is only true if the commutator commutes with both the operators A and B that means in case of . It can be directly proved (Messiah , p. 442; see for this proof also Louisell , p. 137) without using the general Baker-Campbell-Hausdorff-Dynkin formula which last is quite complicated.
We apply now the theorem (B1) to the special case and with t considered as a parameter. For the commutator we find
From this follows
Therefore, the series in (B1) truncates after the sum term to and we find
From this follows
and, more generally, for an arbitrary (smooth) function
but only in few cases one may easily disentangle the operator function on the right-hand side (i.e. separate into products of functions of z and of ).
The operator disentanglement on the right-hand side of (B6) in case of the exponential function is true according to the theorem (B2) from which we conclude in special case
since the commutator commutes with the operators z and .
On the right-hand side of the operator identity (B6) we have separated the basic
Generating function for Hermite polynomials . This operator identity can be represented equivalently
On this occasion we mention that from (B5) with follows the operator identity
where in addition we used the operator identity (8.7). Applied to the function it generates the Hermite polynomials as we repeatedly used it. For an arbitrary function with Taylor series at follows from (B10) by reordering of the sum terms in the arising double sum the operator identity
The inner sum is of the form of Generating functions for Hermite polynomials (without powers of a parameter) and cannot be evaluated in closed way in general case.
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