On employing linear algebra approach to hybrid Sheffer polynomials

1.   Introduction and preliminaries

Special polynomials with two variables are critical from the standpoint of applications. These polynomials make it simple to derive a variety of useful identities and aid in the introduction of new families of special polynomials. Bretti et al. ^[5], established the general classes of two-variable Appell polynomials in utilising the features of an iterated isomorphism linked to Laguerre-type exponentials. Several writers investigated the two variable forms of the Hermite, Laguerre, and truncated exponential polynomials, as well as their generalisations (see ^[3,6,7,8].

The truncated exponential polynomials (TEP) $e_m(u)$ possess the series^[1]:

Many issues in optics and quantum physics use these polynomials. The features of these polynomials aren't well understood. The literatures of Dattoli et al. ^[7] provided the first detailed research of some properties of these polynomials.

The Succeeding series formulation can be used to deduce the most notable features of these polynomials, and the integral illustration for $e_m(u)$ is given as:

which being a significant outcome of relation ^[1]:

Consequently, $e_m(u)$ are defined [7,p. 596 (4)]:

Dattoli et al. ^[7] developed a 2-variable extension of the truncated exponential polynomials(TEP), which is being useful in the evaluation of integrals involving products of special functions and in a variety of optics and quantum mechanics situations.

We recall that the following generating relation ^[7] for 2-variable TEP is given by

and the succeeding series relation

Recalling the generating relation of higher order 2-variable TEP [7,p. 599 (31)]

and the succeeding series relation:

In light of the expressions (1.7), (1.5) and (1.4), we find

Note down that

where $U_m(v)$ denotes the second-order Chebyshev polynomials given by the generating relation ^[1]

We know that the class of Sheffer sequences ^[14] is an important class which appears in a variety of problems in applied mathematics, theoretical physics, approximation theory, and other disciplines of mathematics.

and

$s_{m}(u)$ is determined uniquely in Roman ^[12] by two formal power series and it is given by exponential generating relation for $s_{m}(u)$ as

where the compositional inverse of $f(t)$ is $f^{-1}(t)$.

It should be noted that for $\textsf{g}(t) = 1$, the $s_{m}(u)$ reduces to $\mathfrak{s}_{m}(u)$ called associated-Sheffer sequence and for $f(t) = t$, it reduces to $A_{m}(u)$ known as the Appell sequence ^[2].

These sequences are given by the generating relations

and

respectively.

In the table below, chosen members of the Sheffer, associated Sheffer, and Appell polynomial families are listed:

The truncated exponential-Sheffer polynomials are introduced and studied by Khan et al. ^[9] in 2014. Recalling that the truncated exponential-Sheffer polynomials (TESP) ${}_{e^{(r)}}s_m(u, v)$ are defined by the following generating function:

where $f^{-1}(t)$ is the compositional inverse of $f(t)$.

Note. It should be noted that for $\textsf{g}(t) = 1$, the truncated exponential-Sheffer sequence ${}_{e^{(r)}}s_m(u, v)$ becomes the truncated exponential-associated Sheffer sequence ${}_{e^{(r)}}\mathfrak{s}_m(u, v)$ and for $f(t) = t$, it becomes the truncated exponential-Appell sequence ${}_{e^{(r)}}A_m(u, v)$.

We review some definitions and concepts related to the Pascal and Wronskian matrices which will be used for derivation of the results in Sections 2, 3 and 5.

Let $\{ j(u) = \sum\limits_{k = 0}^\infty a_k \frac{t^i}{i!} = \mathcal{F} \; |\; a_i\; \in \mathbb{C} \}$ be power series in the $\mathbb{C}$-algebra. The generalized Pascal functional matrix ^[15] of an analytic function $\mathrm{g}(t)$ for $\mathrm{g}(t) \in \mathcal{F}$ is a square matrix of order $(m+1)$ denoted by $\mathbb{P}_{m}[\mathrm{g}(t)]$ and defined as:

It should be noted that $\mathrm{g}^{(i)}$ represents the $i^{th}$ order derivative of $\mathrm{g}$ and $\mathrm{g}^{i}$ denotes the $i^{th}$ power of $\mathrm{g}$ throughout the article.

Also, Wronskian matrix of the $n^{th}$ order of an analytic functions $\mathrm{g}_{1}(t), \mathrm{g}_{2}(t), \mathrm{g}_{3}(t), \cdots, \mathrm{g}_{m}(t)$ is an $(m+1)\times m$ matrix defined by:

It is important to note that in the expressions $\mathbb{P}_{m}[\mathrm{g}(x, t)]_{t = 0}$ and $\mathcal{W}_{m}[\mathrm{g}(x, t)]_{t = 0}$, we consider $t$ as working variable and $x$ is as a parameter.

We recall certain important properties and relationships between the Pascal functional and Wronskian matrices ^[16].

For any $a, b \in \mathbb{C}$ and any analytic functions $\mathrm{g}(t), g(t) \in \mathcal{F}$, the following properties hold true:

where $\Lambda_{m} =$ diag$[0!, 1!, 2!, \cdots, m!]$; $\mathrm{g}(0) = 0$ and $\mathrm{g}'(0)\neq 0$.

Further, for any analytic functions $\mathrm{g}_{1}(t), \mathrm{g}_{2}(t), \cdots, \mathrm{g}_{m}(t)$ and $f(t)$, the following property holds true:

The significance of the two variable forms of special polynomials in applications, as well as recent work on Sheffer sequences using a matrix approach ^[10,16], provides an incentive to develop recursive formulas and differential equations for truncated exponential-Sheffer polynomials using the generalised Pascal functional matrix of an analytic function and the Wronskian matrix of several analytic functions. Section 2 establishes several recursive formulas for truncated exponential-Sheffer polynomials ${}_{e^{(r)}}s_m(u, v)$. The differential equations for these polynomials are derived in Section 3. In Section 4, we look at few cases that provide solutions for some hybrid special polynomials. The identity for the truncated exponential-Sheffer polynomial sequences is determined in the final section.

2.   Recursive formulas

To utilize the Wronskian matrices, the vector form of the TESP is required.

The TES vector denoted by ${\bf \overline{{}_{e^{(r)}}s_m}}(u, v)$ is defined as:

where $\{{}_{e^{(r)}}s_m(u, v)\}$ is the TESP sequence defined by Eq (1.17).

Since $\frac{\exp(uf^{-1}(t))}{\textsf{g}(f^{-1}(t))(1-v(f^{-1}(t))^r)}$ is analytic, therefore by Taylor's expansion, it follows that

In view of Eq (2.2), the truncated exponential-Sheffer vector (2.1) can be expressed as:

The following Lemma must be proved before proceeding with the formulation of the recursive formulas for the truncated exponential-Sheffer polynomial sequence.

Lemma 2.1. The following property holds for the truncated exponential-Sheffer polynomial sequence ${}_{e^{(r)}}s_m(u, v)$:

Proof. It follows from using property (1.24) in the r.h.s. of Eq (2.3) that

For, $\mathcal{W}_{m}[e^{xt}]_{t = 0} = [1\; x\; x^{2}\; \cdots\; x^{n}]^{T}$ and using relation (1.23), the above expression can be written as

Division by $k!$ and differentiation of (2.6) $k$ times with regard to $x$ results in

The left part of (2.7) is the $k^{th}$ column of

and the right part of (2.7) is the $k^{th}$ column of

Thus (2.4) is proved.

Next, for ${}_{e^{(r)}}s_m(u, v)$ certain recursive formulas are established.

First, we expresses ${}_{e^{(r)}}s_{m+1}(u, v)$ in terms of ${}_{e^{(r)}}s_m(u, v)$ by deriving a recursive formula and its derivatives in the following manner.

Theorem 2.2. For ${}_{e^{(r)}}s_m(u, v)$, the succeeding recursive formula holds:

where

and

Proof. In light of (1.19) and (2.2), we find

Using expressions (1.22)–(1.24) in a suitable manner and differentiating $\mathcal{W}_{m}\left[\frac{d}{dt}\left(\frac{\exp(uf^{-1}(t))}{\textsf{g}(f^{-1}(t))(1-v(f^{-1}(t))^r)}\right)\right]_{t = 0}$, we find

which in light of Lemma 2.1 becomes

Assertion (2.8) is obtained by comparing the last rows of (2.9) and (2.11).

Remark 2.3. For ${g}(t) = 1 \implies$ $\eta_{k} = 0$ $(k\ge0)$, thus for ${g}(t) = 1$ and as a result of Theorem 2.1, we arrive at the following conclusion.

Corollary 2.4. For ${}_{e^{(r)}}{\mathfrak{s}}_{n}(u, v)$, the succeeding recursive relation holds

where

Remark 2.5. Consequently for $f(t) = t \implies$ $\delta_{0} = 1$; $\delta_{k} = 0$ $(k\neq 0)$, thus for $t = f(t)$ as a result of Theorem 2.1, we arrive at the following conclusion.

Corollary 2.6. For the ${}_{e^{(r)}}A_{m}(u, v)$, the succeeding recursive formula holds

where

Further, a pure recursive formula for the truncated exponential-Sheffer polynomials is derived by proving the result.

Theorem 2.7. For ${}_{e^{(r)}}s_{m}(u, v)$, the succeeding recursive formula holds.

where

Proof. In light of property (1.23), the expression $\mathcal{W}_{m}\left[f'(f^{-1}(t))\frac{d}{dt}\left(\frac{\exp(uf^{-1}(t))}{\textsf{g}(f^{-1}(t))(1-v(f^{-1}(t))^r)}\right)\right]_{t = 0}$ takes the form:

When differentiation is performed in the same expression and properties (1.21) and (1.23) are used, it follows that

Assertion (2.14) is established by comparing the last rows of (2.15) and (2.13).

Remark 2.8. Since ${g}(t) = 1 \implies$ $\theta_{k} = 0$ $(k\ge0)$, thus for ${g}(t) = 1$ and as a result of Theorem 2.2, we arrive at the following conclusion.

Corollary 2.9. For ${}_{e^{(r)}}{\mathfrak{s}}_{n}(u, v),$ the succeeding pure recursive formula holds.

where

Remark 2.10. Since $f(t) = t \implies$ $\epsilon_{0} = 1$; $\epsilon_{k} = 0$ $(k\neq 0)$, thus for $f(t) = t$.

Corollary 2.11. For ${{}_{e^{(r)}}}A_{m}(u, v),$ the succeeding pure recursive formula holds

where

Finally, by proving the following result, we derive a pure recursive formula that provides a representation of ${}_{e^{(r)}}s_{m+1}(u, v)$ in terms of ${}_{e^{(r)}}s_{n-k}(u, v)$ ($k = 0, 1, 2, cdots n$).

Theorem 2.12. For ${}_{e^{(r)}}s_{m}(u, v)$, the succeeding pure recursive formula holds:

where

Proof. Differentiating $\mathcal{W}_{m}\left[\frac{d}{dt}\left(\frac{\exp(uf^{-1}(t))}{\textsf{g}(f^{-1}(t))(1-v(f^{-1}(t))^r)}\right)\right]_{t = 0}$ and then using (1.23), we have

Assertion (2.19) is established by comparing the last rows of (2.9) and (2.20).

Remark 2.13. Since ${g}(t) = 1 \implies$ $\psi_{k} = 0$ $(k\ge0)$, therefore for ${g}(t) = 1$and as a result of Theorem 2.3, we arrive at the following conclusion.

Corollary 2.14. For $_L{\mathfrak{s}}_{n}(u, v)$, the succeeding pure recursive formula holds

where

Remark 2.15. Noting down for $f(t) = t\implies \phi_{0} = 1$; $\phi_{k} = 0$ $(k\neq 0)$. Thus, taking $f(t) = t$, in Theorem 2.3, we deduce the consequence of Theorem 2.3.

Corollary 2.16. For ${}_{e^{(r)}}A_{m}(u, v)$, the succeeding pure recursive formula holds

where

The differential equation satisfied by the truncated exponential-Sheffer polynomials ${}_{e^{(r)}}s_{m}(u, v)$ is derived in the following section.

3.   Differential equation

We prove the following result in order to derive the differential equation for the truncated exponential-Sheffer polynomials ${}_{e^{(r)}}s_{m}(u, v)$.

Theorem 3.1. The following differential equation is satisfied by the truncated exponential-Sheffer polynomials ${}_{e^{(r)}}s_{m}(u, v)$.

where

Proof. In light of (1.23), the relation ${\mathcal{W}}_{n}\left[t\frac{d}{dt}\left(\frac{\exp(uf^{-1}(t))}{\textsf{g}(f^{-1}(t))(1-v(f^{-1}(t))^r)}\right)\right]_{t = 0}$ takes the form

On the other hand, by performing differentiation in the same expression and appropriately applying properties (1.23)–(1.25), we have

In light of Lemma 2.1, (3.3) becomes

Assertion (3.1) is proved by comparing the last rows of (3.2) and (3.4) and in light of the fact that

Remark 3.2. For ${g}(t) = 1$, the polynomials ${}_{e^{(r)}}s_{m}(u, v)$ reduces to the polynomials ${}_{e^{(r)}}{\mathfrak{s}}_{n}(u, v)$ and since ${g}(t) = 1 \implies \alpha_{k} = 0 \; (k\ge 1)$, the succeeding consequence of Theorem 3.1 is obtained.

Corollary 3.3. The polynomials ${}_{e^{(r)}}{\mathfrak{s}}_{n}(u, v)$ satisfy the succeeding differential equation

where

Remark 3.4. For $f(t) = t$, the polynomials ${}_{e^{(r)}}s_{m}(u, v)$ reduces to the polynomials ${}_{e^{(r)}}A_{m}(u, v)$ and since $f(t) = t \implies \beta_{1} = 1$; $\beta_{k} = 0 \; (k\neq 1)$, therefore, for $f(t) = t$, the succeeding consequence of Theorem 3.1 is obtained.

Corollary 3.5. The polynomials ${}_{e^{(r)}}A_{m}(u, v)$ satisfy the succeeding differential equation

where

The differential equation and recursive formulas for certain members of the truncated exponential-Sheffer and truncated exponential-associated Sheffer families are derived in the following section.

4.   Examples

We use Theorem 3.1 to derive the differential equation and Theorems 2.1–2.3 to find the recursive formulas for certain members of the truncated exponential-Sheffer family.

Example 4.1, For $e^{(\frac{t}{\nu})^m}, \; f(t) = \frac{t}{\nu} = {g}(t)$ and $\nu t = f^{-1}(t)$, the Sheffer polynomials become the generalized Hermite polynomials $H_{n, m, \nu}(x)$ and consequently the truncated exponential-Sheffer polynomials become the truncated exponential-generalized Hermite polynomials ${}_{e^{(r)}}H_{n, m, \nu}(u, v).$

It folllows from Theorem 3.1 that

Inserting (4.1) into expression (3.1), the differential equation for the truncated exponential-generalized Hermite polynomials ${}_{e^{(r)}}H_{n, m, \nu}(u, v)$ is obtained as follows

Similarly, in light of Theorems 2.1–2.3, the recursive formulas for the truncated exponential-generalized Hermite polynomials ${}_{e^{(r)}}H_{n, m, \nu}(u, v)$ are obtained as follows

and

Example 4.2. For $(1-t)^{-\kappa-1} = {g}(t)$ and $\frac{t}{t-1} = f^{-1}(t)$, the Sheffer polynomials become the generalized Laguerre polynomials $L^{\kappa}_n(x)$ and consequently the truncated exponential-Sheffer polynomials become the generalized truncated exponential-Laguerre polynomials ${}_{e^{(r)}}L^{\kappa}_n(u, v).$

From Theorem 3.1, we find

Inserting (4.7) into (3.1), the succeeding differential equation for ${}_{e^{(r)}}L^{\kappa}_n(u, v)$ is obtained

Similarly, in light of Theorems 2.1–2.3, the succeeding recursive formulas for ${}_{e^{(r)}}L^{\kappa}_n(u, v)$ are obtained

Next, The differential equation is then derived using Corollary 3.1, and the recursive formulas for certain members of the truncated exponential-associated Sheffer family are derived using Corollaries 2.1, 2.3, and 2.5.

Example 4.3. For $\ln(1+t) = f(t)$ and $e^t -1 = f^{-1}(t)$, the associated Sheffer polynomials reduces to the exponential polynomials $\phi_{n}(x)$ and consequently the truncated exponential-associated Sheffer polynomials become the truncated exponential-exponential polynomials ${}_{e^{(r)}}\phi_n(u, v)$.

It follows from Corollary 3.1

Inserting (4.13) into (3.5), the succeeding differential equation for polynomials ${}_{e^{(r)}}\phi_{n}(u, v)$ is obtained:

Similarly, Corollaries 2.1, 2.3, and 2.5 are used to obtain the recursive formulas for the polynomials ${}_{e^{(r)}}\phi_n(u, v)$

and

In the following section, we look at an identity for truncated exponential-Sheffer polynomials.

5.   Appendix

The approach used in the preceding sections can be extended to yield additional results. As an example, we prove the following result to derive an identity for the truncated exponential-Sheffer polynomial sequences.

Theorem 5.1. The following identity for the truncated exponential-Sheffer polynomials ${}_{e^{(r)}}s_m(u, v)$ holds true

where $f^{(k)}(0) = \Lambda_k.$

Proof. In light of property (1.23), expression $\mathcal{W}_m\left[t \left(\frac {\exp(uf^{-1}(t))}{g(f^{-1}(t))(1-v(f^{-1}(t))^r)}\right) \right]_{t = 0}$ takes the the form as given under

In contrast, using Lemma 2.1 and the properties (1.22)–(1.24), we have

Assertion (5.1) is proved by comparing the $n^{th}$ rows of (5.2) and (5.3) and replacing $m$ by $m+1$.

To demonstrate the application of Theorem 5.1, consider the following examples in the form of a table.

The recursive formulas and differential equations are critical in the analysis of algorithms ^[13]. These appear in infinite impulse response (IIR) digital filters. The attempt to model population dynamics gave rise to recurrence relations. Fibonacci numbers, for example, were once used as a model for rabbit population growth.

6.   Conclusions

Hence we provided some specific properties of truncated exponential-Sheffer polynomials and multi variable special polynomial properties. The properties and association between the Pascal functional and Wronskian matrices are used to build the recursive equations and differential equation for these polynomials, as well as for several members of the truncated exponential-Sheffer family. The corresponding results for the truncated exponential-associated Sheffer and truncated exponential-Appell families are provided with some examples.

Conflict of interest

The author declares that there is no conflict of interest.