(<i>p</i>,<i>q</i>)‐Sturm‐Liouville problems and their orthogonal solutions

In this note, for the (p, q)-derivative operator $$D_{p, q}$$ D p , q defined by $$\begin{aligned}D_{p, q}f(x)={\left\{ \begin{array}{ll} \frac{f(px)-f(qx)}{(p-q)x} & \text {if } x\ne 0;\\ f'(0), & \text {if}~x=0. \end{array}\right. }, \quad \text { where } p\ne q,\end{aligned}$$ D p , q f ( x ) = f ( p x ) - f ( q x ) ( p - q ) x if x ≠ 0 ; f ′ ( 0 ) , if x = 0 . , where p ≠ q , we investigate the following property: $$\begin{aligned}(D_{p, q}^{n}f)(0)= & \lim _{x\rightarrow 0}D_{p, q}^{n}f(x)=\frac{f^{(n)}(0)((p, q); (p, q))_{n}}{n!(p-q)^n}\\= & \frac{f^{(n)}(0)[n]_{p, q}!}{n!}, \quad n\in \{1, 2, \ldots \},\end{aligned}$$ ( D p , q n f ) ( 0 ) = lim x → 0 D p , q n f ( x ) = f ( n ) ( 0 ) ( ( p , q ) ; ( p , q ) ) n n ! ( p - q ) n = f ( n ) ( 0 ) [ n ] p , q ! n ! , n ∈ { 1 , 2 , … } , for which $$f^{(n)}(0)$$ f ( n ) ( 0 ) exists. The real variable and the complex variable cases are considered. Specializing to the case $$p=1$$ p = 1 , we recover the property for the q-derivative operator as obtained by J. Koekoek and R. Koekoek in 1993, in related paper. Furthermore, some applications were discussed.

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A note on the post quantum-Sheffer polynomial sequences

Khan,

Haneef

2024

Forum Mathematicum

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In this article, the post quantum analogue of Sheffer polynomial sequences is introduced using concepts of post quantum calculus. The series representation, recurrence relations, determinant expression and certain other properties of this class are established. Further, the 2D-post quantum-Sheffer polynomials are introduced via generating function and their properties are established. Certain identities and integral representations for the 2D-post quantum-Hermite polynomials, 2D-post quantum-Laguerre polynomials, and 2D-post quantum-Bessel polynomials are also considered.

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(p,q)‐Sturm‐Liouville problems and their orthogonal solutions

Cited by 3 publications

References 27 publications

Generating Functions of (p, q)-Analogue of I-Function Satisfying Truesdell’s Ascending and Descending $$F_{p,q}$$-Equation

Generating Functions of (p, q)-Analogue of I-Function Satisfying Truesdell’s Ascending and Descending $$F_{p,q}$$-Equation

A Note on the (p, q)-Derivative Operator

A note on the post quantum-Sheffer polynomial sequences

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