New developments for the Jacobi polynomials

Duriye Korkmaz Duzgun

doi:10.1515/dema-2025-0157

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New developments for the Jacobi polynomials

Published/Copyright: October 7, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

In this work, first, a new and more general form of the Jacobi differential equation is developed, and the k -Jacobi polynomials are defined by means of the general solution of this equation and related generating functions and Rodrigues formula are obtained. Its orthogonality is also shown and its norm is derived. Subsequently, properties similar to those of the k-Jacobi polynomials are achieved by defining the k-Gegenbauer and k-Legendre differential equations and the k-Gegenbauer and k-Legendre polynomials corresponding to a special solution of them. These polynomials also have several new properties, including explicit formulas, generating functions, and recurrence relations. In addition, a certain class of bilateral and bilinear generating functions are derived and some examples are presented.

Keywords: Jacobi polynomials; Legendre polynomials; hypergeometric function; Rodrigues formula; orthogonal polynomials; Pochhammer symbol

MSC 2020: 33C05; 33C45; 33B15

1 Introduction

Jacobi introduced Jacobi polynomials in relation to the solution of a Gauss hypergeometric equation known as the Jacobi differential equation [1]. Later, other generalized Jacobi polynomials were defined, which in certain cases are transformed into these polynomials. A few examples follow: Andrews et al. investigated the q-Jacobi polynomials obtained using q-analysis in [2]; Ismail and Masson described a family of orthogonal polynomials, which are a general form of Jacobi polynomials [3]; He and Natalini introduced polynomials that produce Jacobi polynomials when N → ∞ and are known as relativistic Jacobi polynomials [4]; Koekoek and Koekoek investigated polynomials P n ( α , β , M , N ) ( x ) , a generalization of the Jacobi polynomials, as a polynomial solution to a second order differential equation [5]; Grünbaum developed the Jacobi polynomials with matrix variables [6]; Guo et al. studied generalized Jacobi polynomials with indexes α or β ≤ − 1 [7]; and Arfaoi and Ben-Mabrouk researched the Clifford-Jacobi polynomials using Clifford analysis [8].

Diaz and Pariguan suggested k -generalization of the Pochhammer symbol in [9]. Following the definition of the k -gamma and k -beta functions in [9], research studies have been conducted on k -generalizations for Bessel [10], Mittag-Leffler [11], Struve [12], Wright [13], Meijer’s G [14], and Airy functions [15]. In addition, there are limited studies on polynomial extensions [16,17] on subjects such as certain features of the k -gamma function and the k -hypergeometric function, k -fractional derivative operators [18], and certain forms of inequalities [19]. Among the family of orthogonal polynomials, only the k-generalization and properties of Laguerre polynomials have been studied (see [17]).

In this article, k -generalization of Jacobi polynomials is presented, which has an important place in applied mathematics and is related to various orthogonal polynomials, as a polynomial solution of a k -hypergeometric equation.

Here are some definitions that will be used throughout the research. The study of k -generalizations in special function theory started with the definition of the k -Pochhammer symbol by Pariguan and Diaz [9]. For k ∈ R , the k -Pochhammer symbol is defined by

(1) ( c ) n , k = c ( c + k ) ( c + 2 k ) ( c + 3 k ) … ( c + ( n − 1 ) k ) ,

and c ∈ C , n ∈ N , ( c ) 0 , k = 1 [9].

The k -gamma function is

(2) Γ k ( x ) = ∫ 0 ∞ t x − 1 e − t k k d t ,

where k > 0 and Re ( x ) > 0 [9]. The relation between ( c ) n , k and Γ k ( x ) is identified as

(3) ( c ) n , k = Γ k ( c + n k ) Γ k ( c ) ,

which is given in [9].

The k -beta function is

(4) B k ( x , y ) = 1 k ∫ 0 1 t x k − 1 ( 1 − t ) y k − 1 d t ,

where k > 0 and Re ( x ) > 0 , Re ( y ) > 0 [9]. In terms of the k -gamma function, the k -beta function is

B k ( x , y ) = Γ k ( x ) Γ k ( y ) Γ k ( x + y ) ,

which is equivalent to (4), [9].

The k -generalized hypergeometric series has the following definition:

(5) F q , k p a 1 , … , a p ; z b 1 , … , b q ; ≔ 1 + ∑ n = 1 ∞ ∏ i = 1 p ( a i ) n , k ∏ j = 1 q ( b j ) n , k z n n ! ,

where k > 0 and p , q ∈ N 0 = { 0 , 1 , 2 , … } are any number of numerator and denominator parameters [20]. In (5), if we take p = 0 , q = 1 , we have

(6) F 1 , k 0 ( − ; b ; z ) = ∑ n = 0 ∞ 1 ( b ) n , k n ! z n ,

and if we take q = 1 , p = 2 , we obtain the k -hypergeometric function as (see [21])

(7) F 1 , k 2 ( a , b ; c ; z ) = F k ( a , b ; c ; z ) = ∑ n = 0 ∞ ( a ) n , k ( b ) n , k ( c ) n , k n ! z n ,

that is provided by the k -hypergeometric differential equation (see [21])

(8) k z ( 1 − k z ) y ″ + [ c − ( a + b + k ) k z ] y ′ − a b y = 0 .

The general solution of (8) has the following form [22]:

(9) y ( z ) = A F k ( a , b ; c ; z ) + B z 1 − c k F k ( a − c + k , b − c + k ; 2 k − c ; z ) .

If we take k = 1 in equations (1), (2), (4), (5), (7), and (8), we obtain the known Pochhammer symbol, the gamma function, the beta function, the generalized hypergeometric function, the hypergeometric function, and the hypergeometric equation, respectively.

The structure of the research is as follows: In Section 2, the k -Jacobi differential equation is defined. The polynomial solution is obtained by transforming this equation into the k -hypergeometric equation. Then, k -Jacobi polynomials are defined. The explicit formula is obtained on the basis of this definition of k -Jacobi polynomials. Two different generating function relations are obtained for k -Jacobi polynomials. Furthermore, the Rodrigues formula is found for the k -Jacobi polynomials. Then, its orthogonality is examined, and its norm is determined. In Section 3, the k -Gegenbauer polynomials are defined by taking α = β = ν − 1 ⁄ 2 in the definition of the k -Jacobi polynomials and some similar properties are given. All the results in Section 2 are also obtained for k -Legendre polynomials in Section 4 by taking α = β = 0 in the definition of the k -Jacobi polynomials. Additionally, new explicit formula, generating functions, and recurrence relations are derived for these polynomials. In Section 5, several bilateral and bilinear generating functions are explored for these polynomials and also some examples are displayed. The last section of the article is devoted to the conclusions.

2 k -Jacobi polynomials

First, the k -Jacobi differential equation is defined, and then the problem to find the polynomial solution is solved. The k -Jacobi polynomials are also derived by using this solution. After providing the generating function relation, explicit formula, and Rodrigues formula, its orthogonality is demonstrated, and its norm is determined.

Definition 2.1

We introduce the k -Jacobi differential equation as follows:

(10) k ( 1 − x 2 ) y ″ + [ β − α − ( α + β + k + 1 ) x ] y ′ + n ( n k + α + β + 1 ) y = 0 ,

where k > 0 .

Theorem 2.1

The general solution of the k-Jacobi differential equation given by (10) is

(11) y ( x ) = A F k − n k , n k + α + β + 1 ; 2 α + k + 1 2 ; 1 − x 2 k + B 1 − x 2 k k − 2 α − 1 2 × F k − 2 n k − 2 α + k − 1 2 , 2 n k + 2 β + k + 1 2 ; 3 k − 1 − 2 α 2 ; 1 − x 2 k ,

where A and B are arbitrary constants.

Proof

If we substitute x = 1 − 2 k u , in (10), we obtain the following k -hypergeometric differential equation:

(12) k u ( 1 − k u ) d 2 y d u 2 + 2 α + k + 1 2 − ( α + β + k + 1 ) k u d y d u + n k ( n k + α + β + 1 ) y = 0 .

Considering (8) and (9), we obtain the solution of the k -differential equation (12) as

(13) y ( u ) = A F k − n k , n k + α + β + 1 ; 2 α + k + 1 2 ; u + B u k − 2 α − 1 2 F k − 2 n k − 2 α + k − 1 2 , 2 n k + 2 β + k + 1 2 ; 3 k − 1 − 2 α 2 ; u .

When we write u = 1 − x 2 k in (13), we reach the solution of equation (10), and the proof is completed.□

We can use the first part of the solution (11) to define the k -Jacobi polynomials. This part can be written as

y ( x ) = A ∑ r = 0 ∞ ( − n k ) r , k ( n k + α + β + 1 ) r , k 2 α + k + 1 2 r , k r ! 1 − x 2 k r .

Here, from the definition of the k -Pochhammer symbol, ( − n k ) r , k = 0 for all values with r > n . So, the series terminates and the hypergeometric function turns into a hypergeometric polynomial and is written as

(14) y ( x ) = A ∑ r = 0 n ( − n k ) r , k ( n k + α + β + 1 ) r , k 2 α + k + 1 2 r , k r ! 1 − x 2 k r .

By selecting

A = ( 1 + α ) n , k n !

in (14), we can define a n th order polynomial that satisfies the k -Jacobi differential equation given by (10). Thus, we can now define the k -Jacobi polynomials as follows:

Definition 2.2

The k -Jacobi polynomials P n , k ( α , β ) ( x ) are expressed as

(15) P n , k ( α , β ) ( x ) = ( 1 + α ) n , k n ! F k − n k , n k + α + β + 1 ; 2 α + k + 1 2 ; 1 − x 2 k .

Equation (15) allows us to write

P 0 , k ( α , β ) ( x ) = 1 , P 1 , k ( α , β ) ( x ) = ( 1 + α ) + 2 k ( 1 + α ) ( α + β + k + 1 ) ( 2 α + k + 1 ) x − 1 2 k , P 2 , k ( α , β ) ( x ) = ( 1 + α ) ( 1 + α + k ) 2 + 2 k ( 1 + α ) ( 1 + α + k ) ( α + β + 2 k + 1 ) ( 2 α + k + 1 ) x − 1 2 k + 2 k 2 ( 1 + α ) ( 1 + α + k ) ( α + β + 2 k + 1 ) ( α + β + 3 k + 1 ) ( 2 α + k + 1 ) ( 2 α + 3 k + 1 ) x − 1 2 k 2 .

Lemma 2.1

The k-Pochhammer symbol verifies the following equality:

(16) ( k − c − n k ) m , k = ( c ) n , k ( − 1 ) m ( c ) n − m , k .

Proof

By applying the k -Pochhammer symbol definition, we obtain

( k − c − n k ) m , k = ( k − c − n k ) ( k − c − n k + k ) … ( k − c − n k + m k − k ) = 1 ( − 1 ) m ( c + n k − k ) ( c + n k − 2 k ) … ( c + n k − m k ) = ( c + n k − k ) … ( c + n k − m k ) ( c + n k − m k − k ) … ( c + k ) c ( − 1 ) m ( c + n k − m k − k ) … ( c + k ) c = ( c ) n , k ( − 1 ) m ( c ) n − m , k ,

which completes the proof.□

Theorem 2.2

The k-Jacobi polynomials have the following explicit formula:

(17) P n , k ( α , β ) ( x ) = ∑ r = 0 n ( 1 + α ) n , k 2 β + k + 1 2 n , k k n x − 1 2 k r x + 1 2 k n − r 2 α + k + 1 2 r , k 2 β + k + 1 2 n − r , k r ! ( n − r ) ! .

Proof

From symmetry of the parameters in the numerator of the k -hypergeometric function, by using in (15) the transformation formula [23]

F k ( a , b ; c ; z ) = ( 1 − k z ) − b k F k c − a , b ; c ; − z 1 − k z

and taking a = n k + α + β + 1 , b = − n k , c = 2 α + k + 1 2 , and z → 1 − x 2 k in (15), we obtain

(18) P n , k ( α , β ) ( x ) = ( 1 + α ) n , k n ! 1 + x 2 n F k − 2 β + k − 1 − 2 n k 2 , − n k ; x − 1 k ( x + 1 ) 2 α + k + 1 2 ; .

Equation (18) is a hypergeometric form different from (15) for k -Jacobi polynomials. In this way, we can write

(19) P n , k ( α , β ) ( x ) = ∑ r = 0 n ( 1 + α ) n , k − 2 β + k − 1 − 2 n k 2 r , k ( − n k ) r , k 1 + x 2 n x − 1 x + 1 r 2 α + k + 1 2 r , k k r n ! r ! .

Using the identity (16) with c = 2 β + k + 1 2 , m = r in the Lemma 2.1, we have

(20) − 2 β + k − 1 − 2 n k 2 r , k = 2 β + k + 1 2 n , k ( − 1 ) r 2 β + k + 1 2 n − r , k .

Using the identity [24]

( − c ) r , k = k r ( − 1 ) r ( c k ) ! ( c k − r ) ! ,

with c = n k , we deduce

(21) ( − n k ) r , k = k r ( − 1 ) r n ! ( n − r ) ! .

If we use (20) and (21) in (19), we achieve the desired result.□

Theorem 2.3

The k-Jacobi polynomials have the following generating relation:

(22) ∑ n = 0 ∞ ( 1 + α + β ) n , k P n , k ( α , β ) ( x ) ( 1 + α ) n , k t n = ( 1 − k t ) − 1 + α + β k F k 1 + α + β 2 , 1 + α + β + k 2 ; 2 t ( x − 1 ) ( 1 − k t ) 2 2 α + k + 1 2 ; ,

where k > 0 .

Proof

Let T be denoted as the first member of assertion (22). Using (15), we have the following sum:

T = ∑ n = 0 ∞ ∑ r = 0 n ( 1 + α + β + n k ) r , k ( 1 + α + β ) n , k ( − n k ) r , k 1 − x 2 k r t n 2 α + k + 1 2 r , k n ! r ! .

From the formula [25]

(23) ( c ) n + r , k = ( c ) n , k ( c + n k ) r , k = ( c ) r , k ( c + r k ) n , k ,

with c = 1 + α + β in (23), it holds true

(24) ( 1 + α + β ) n , k ( 1 + α + β + n k ) r , k = ( 1 + α + β ) n + r , k .

If we use (21) and (24), we obtain

T = ∑ n = 0 ∞ ∑ r = 0 n ( 1 + α + β ) n + r , k 2 α + k + 1 2 r , k ( n − r ) ! r ! x − 1 2 r t n .

By substituting n + r for n , we obtain

T = ∑ n = 0 ∞ ∑ r = 0 ∞ ( 1 + α + β ) n + 2 r , k 2 α + k + 1 2 r , k n ! r ! x − 1 2 r t n + r .

If we take c = 1 + α + β and r → 2 r in (23), we have

( 1 + α + β ) n + 2 r , k = ( 1 + α + β ) 2 r , k ( 1 + α + β + 2 r k ) n , k .

So, we obtain

T = ∑ n = 0 ∞ ∑ r = 0 ∞ ( 1 + α + β ) 2 r , k ( 1 + α + β + 2 r k ) n , k 2 α + k + 1 2 r , k n ! r ! t ( x − 1 ) 2 r t n = ∑ r = 0 ∞ ( 1 + α + β ) 2 r , k 2 α + k + 1 2 r , k r ! t ( x − 1 ) 2 r ∑ n = 0 ∞ ( 1 + α + β + 2 r k ) n , k n ! t n .

Here, using identity [25]

( c ) 2 r , k = 2 2 r c 2 r , k c 2 + k 2 r , k ,

with c = α + β + 1 , we obtain

( α + β + 1 ) 2 r , k = 2 2 r α + β + 1 2 r , k α + β + 1 + k 2 r , k .

Furthermore, from equality [9]

( 1 − k t ) − c k = ∑ n = 0 ∞ ( c ) n , k n ! t n ,

with c = 1 + α + β + 2 r k , we have

( 1 − k t ) − 1 + α + β k − 2 r = ∑ n = 0 ∞ ( α + β + 1 + 2 r k ) n , k n ! t n .

Then, we reach

T = ( 1 − k t ) − 1 + α + β k ∑ r = 0 ∞ α + β + 1 2 r , k α + β + 1 + k 2 r , k 2 α + k + 1 2 r , k r ! 2 t ( x − 1 ) ( 1 − k t ) 2 r .

If we apply definition (7) to the last equation, we obtain the desired result, and the proof is completed.□

Theorem 2.4

The following Bateman-type generating function relation holds true for the k-Jacobi polynomials:

(25) ∑ n = 0 ∞ P n , k ( α , β ) ( x ) t n ( 1 + α ) n , k 2 β + k + 1 2 n , k = F 1 , k 0 − ; ( x + 1 ) t 2 2 β + k + 1 2 ; F 1 , k 0 − ; ( x − 1 ) t 2 2 α + k + 1 2 ; ,

where k > 0 .

Proof

After multiplying both sides of (17) by t n ( 1 + α ) n , k 2 β + k + 1 2 n , k , summing up from n = 0 to n = ∞ , we obtain

∑ n = 0 ∞ P n , k ( α , β ) ( x ) t n ( 1 + α ) n , k 2 β + k + 1 2 n , k = ∑ n = 0 ∞ ∑ r = 0 n k n x − 1 2 k r x + 1 2 k n − r t n 2 α + k + 1 2 r , k 2 β + k + 1 2 n − r , k r ! ( n − r ) ! = ∑ n = 0 ∞ ( x + 1 ) t 2 n 2 β + k + 1 2 n , k n ! ∑ r = 0 ∞ ( x − 1 ) t 2 r 2 α + k + 1 2 r , k r ! .

The desired result is obtained from the definition of the k -hypergeometric series (6).□

Theorem 2.5

The following Rodrigues formula holds true for the k-Jacobi polynomials:

(26) P n , k ( α , β ) ( x ) = ( 1 + α ) n , k k n ( x − 1 ) k − 2 α − 1 2 k ( x + 1 ) k − 2 β − 1 2 k 2 α + k + 1 2 n , k 2 n n ! D n ( x − 1 ) n + 2 α + 1 − k 2 k ( x + 1 ) n + 2 β + 1 − k 2 k ,

where D = d d x is the derivative operator.

Proof

It is known that for the non-negative integers m and s ,

D s x m + α k = m + α k m + α k − 1 … m + α k − s + 1 x m − s + α k .

If we multiply and divide the right side of this last expression by m + α k − s … 2 + α k 1 + α k , we have

D s x m + α k = m + α k m + α k − 1 … m + α k − s + 1 ⋅ m + α k − s … 2 + α k 1 + α k m + α k − s … 2 + α k 1 + α k x m − s + α k = ( α + k ) … ( α + m k ) k m − s ( α + k ) … ( α + m k − s k ) k m x m − s + α k .

From the k -Pochhammer symbol, we have

D s x m + α k = ( α + k ) m , k k s ( α + k ) m − s , k x m − s + α k ,

D r ( x + 1 ) n + 2 β + 1 − k 2 k = 2 β + k + 1 2 n , k k r 2 β + k + 1 2 n − r , k ( x + 1 ) n − r + 2 β + 1 − k 2 k ,

and

D n − r ( x − 1 ) n + 2 α + 1 − k 2 k = 2 α + k + 1 2 n , k k n − r 2 α + k + 1 2 r , k ( x − 1 ) r + 2 α + 1 − k 2 k .

Rearranging (17) in the light of the above results and using the Leibnitz theorem, one obtains

P n , k ( α , β ) ( x ) = ∑ r = 0 n ( 1 + α ) n , k 2 α + k + 1 2 n , k 2 α + k + 1 2 n , k 2 α + k + 1 2 r , k ( x − 1 ) r 2 β + k + 1 2 n , k 2 β + k + 1 2 n − r , k ( x + 1 ) n − r 1 2 n ( n − r ) ! r ! = ( 1 + α ) n , k k n ( x − 1 ) k − 2 α − 1 2 k ( x + 1 ) k − 2 β − 1 2 k 2 α + k + 1 2 n , k 2 n n ! × ∑ r = 0 n n r D n − r ( x − 1 ) n + 2 α + 1 − k 2 k D r ( x + 1 ) n + 2 β + 1 − k 2 k = ( 1 + α ) n , k k n ( x − 1 ) k − 2 α − 1 2 k ( x + 1 ) k − 1 − 2 β 2 k 2 α + k + 1 2 n , k 2 n n ! D n ( x − 1 ) n + 2 α + 1 − k 2 k ( x + 1 ) n + 2 β + 1 − k 2 k ,

which completes the proof.□

Theorem 2.6

For Re ( α ) > − ( k + 1 ) ⁄ 2 and Re ( β ) > − ( k + 1 ) ⁄ 2 , the k-Jacobi polynomials are orthogonal with respect to the weight function ( 1 − x ) 2 α + 1 − k 2 k ( 1 + x ) 2 β + 1 − k 2 k in the interval ( − 1, 1 ) .

Proof

Since k -Jacobi polynomials satisfy equation (10), we can write

( 1 − x 2 ) D 2 P n , k ( α , β ) ( x ) + β − α k − α + β + k + 1 k x D P n , k ( α , β ) ( x ) + n n k + α + β + 1 k P n , k ( α , β ) ( x ) = 0 .

Since β − α k − α + β + k + 1 k x = 2 β + 1 + k 2 k ( 1 − x ) − 2 α + 1 + k 2 k ( 1 + x ) , we can express this equation in the form

( 1 − x ) 2 α + 1 + k 2 k ( 1 + x ) 2 β + 1 + k 2 k D 2 P n , k ( α , β ) ( x ) + 2 β + 1 + k 2 k ( 1 − x ) − 2 α + 1 + k 2 k ( 1 + x ) ( 1 − x ) 2 α + 1 − k 2 k ( 1 + x ) 2 β + 1 − k 2 k D P n , k ( α , β ) ( x ) + n n k + α + β + 1 k ( 1 − x ) 2 α + 1 − k 2 k ( 1 + x ) 2 β + 1 − k 2 k P n , k ( α , β ) ( x ) = 0 ,

which yields

D ( 1 − x ) 2 α + 1 + k 2 k ( 1 + x ) 2 β + 1 + k 2 k D P n , k ( α , β ) ( x ) + n n k + α + β + 1 k ( 1 − x ) 2 α + 1 − k 2 k ( 1 + x ) 2 β + 1 − k 2 k P n , k ( α , β ) ( x ) = 0 .

Let us first multiply this last equation by P m , k ( α , β ) ( x ) . After changing roles of n and m in the resulting equation and subtracting the two equations side by side, we have

n n k + α + β + 1 k − m m k + α + β + 1 k ( 1 − x ) 2 α + 1 − k 2 k ( 1 + x ) 2 β + 1 − k 2 k P n , k ( α , β ) ( x ) P m , k ( α , β ) ( x ) = D ( 1 − x ) 2 α + 1 + k 2 k ( 1 + x ) 2 β + 1 + k 2 k { P n , k ( α , β ) ( x ) D P m , k ( α , β ) ( x ) − P m , k ( α , β ) ( x ) D P n , k ( α , β ) ( x ) } .

If we integrate both sides in the interval ( − 1 , 1 ) , we obtain the following for Re ( α ) > − ( k + 1 ) ⁄ 2 and Re ( β ) > − ( k + 1 ) ⁄ 2 :

(27) ∫ − 1 1 ( 1 − x ) 2 α + 1 − k 2 k ( 1 + x ) 2 β + 1 − k 2 k P n , k ( α , β ) ( x ) P m , k ( α , β ) ( x ) d x = 0 , m ≠ n ,

which completes the proof.□

Theorem 2.7

The norm of k-Jacobi polynomials ‖ P n , k ( α , β ) ‖ 2 is obtained as

(28) ∫ − 1 1 ( 1 − x ) 2 α + 1 − k 2 k ( 1 + x ) 2 β + 1 − k 2 k [ P n , k ( α , β ) ( x ) ] 2 d x = 2 α + β + 1 k k n + 1 [ ( 1 + α ) n , k ] 2 Γ k 2 α + k + 1 + 2 n k 2 Γ k 2 β + k + 1 + 2 n k 2 n ! 2 α + k + 1 2 n , k 2 ( 1 + α + β + 2 n k ) Γ k ( 1 + α + β + n k ) .

Proof

From (26), we can write

( 1 − x ) 2 α + 1 − k 2 k ( 1 + x ) 2 β + 1 − k 2 k P n , k ( α , β ) ( x ) = ( 1 + α ) n , k k n ( − 1 ) n 2 α + k + 1 2 n , k 2 n n ! D n ( 1 − x ) n + 2 α + 1 − k 2 k ( 1 + x ) n + 2 β + 1 − k 2 k .

If both sides of the last equation are multiplied by P n , k ( α , β ) ( x ) and integrated in the interval ( − 1 , 1 ) for Re ( α ) > − ( k + 1 ) ⁄ 2 , Re ( β ) > − ( k + 1 ) ⁄ 2 , we have

∫ − 1 1 ( 1 − x ) 2 α + 1 − k 2 k ( 1 + x ) 2 β + 1 − k 2 k P n , k ( α , β ) ( x ) P n , k ( α , β ) ( x ) d x = ( 1 + α ) n , k k n ( − 1 ) n 2 α + k + 1 2 n , k 2 n n ! ∫ − 1 1 D n ( 1 − x ) n + 2 α + 1 − k 2 k ( 1 + x ) n + 2 β + 1 − k 2 k P n , k ( α , β ) ( x ) d x .

By applying n times partial integration to the integral on the right-hand side of this equation, we obtain

(29) ∫ − 1 1 ( 1 − x ) 2 α + 1 − k 2 k ( 1 + x ) 2 β + 1 − k 2 k [ P n , k ( α , β ) ( x ) ] 2 d x = ( 1 + α ) n , k k n ( − 1 ) 2 n 2 α + k + 1 2 n , k 2 n n ! ∫ − 1 1 ( 1 − x ) n + 2 α + 1 − k 2 k ( 1 + x ) n + 2 β + 1 − k 2 k [ D n P n , k ( α , β ) ( x ) ] d x .

On the other hand, according to the derivative formula in [26], we obtain

(30) D n P n , k ( α , β ) ( x ) = ( 1 + α ) n , k ( 1 + α + β ) 2 n , k 2 n 2 α + k + 1 2 n , k ( 1 + α + β ) n , k .

Now, substituting (30) in (29) and using (4), after some necessary arrangements, we obtain

∫ − 1 1 ( 1 − x ) 2 α + 1 − k 2 k ( 1 + x ) 2 β + 1 − k 2 k [ P n , k ( α , β ) ( x ) ] 2 d x = 2 α + β + 1 k k n + 1 [ ( 1 + α ) n , k ] 2 ( 1 + α + β ) 2 n , k Γ k 2 α + k + 1 + 2 n k 2 Γ k 2 β + k + 1 + 2 n k 2 n ! 2 α + k + 1 2 n , k 2 ( 1 + α + β ) n , k Γ k ( 1 + α + β + 2 n k + k ) .

Finally, using relation (3) and Γ k ( x + k ) = x Γ k ( x ) (see in [9]), the proof is completed.□

3 k -Gegenbauer polynomials

By means of the k -Jacobi polynomials defined in Section 2, k -expansions of many special functions known in the literature can be defined and their properties can be obtained.

Certain integrals need to be computed when the methods used in mathematical physics to ascertain the quantum mechanical characteristics of systems with many electrons, like molecules, are applied to the molecular system. Many molecular integrals including kinetic energy and electron repulsion can be calculated using Gegenbauer polynomials, which have led to substantially more efficient results when used in electronic structure calculations of molecules [32]. The Gegenbauer polynomials are denoted C n ν ( x ) . These polynomials and Jacobi polynomails have the following relationship (see [27]):

C n ν ( x ) = ( 2 ν ) n ν + 1 2 n P n ( ν − 1 2 , ν − 1 2 ) ( x ) .

In this section, k -Gegenbauer equation and k -Gegenbauer polynomials are defined. Many properties of k -Gegenbauer polynomials can be obtained as special cases of the aforementioned properties of Jacobi polynomials by setting β = α = ν − 1 2 and appealing to the relationship

(31) C n , k ν ( x ) = ( 2 ν ) n , k ν + 1 2 n , k P n , k ( ν − 1 2 , ν − 1 2 ) ( x ) .

Definition 3.1

We introduce the k -Gegenbauer differential equation as follows:

(32) k ( 1 − x 2 ) y ″ − ( 2 ν + k ) x y ′ + n ( 2 ν + n k ) y = 0 ,

where k > 0 .

Theorem 3.1

The general solution of the k-Gegenbauer differential equation given by (32) is

(33) y ( x ) = A F k − n k , n k + 2 ν ; ν + k 2 ; 1 − x 2 k + B 1 − x 2 k 1 2 − ν k F k − n k − ν + k 2 ; n k + ν + k 2 ; 3 k 2 − ν ; 1 − x 2 k .

From the definition of the hypergeometric function in the first part of the solution (33), ( − n k ) r , k = 0 for r > n . Thus, the hypergeometric function turns into a hypergeometric polynomial and by selecting A = ( 2 ν ) n , k n ! , we can define the hypergeometric form of n th order k -Gegenbauer polynomials. This is also found in equality (15), since both α = β = ν − 1 2 .

Definition 3.2

The k -Gegenbauer polynomials C n , k ν ( x ) are defined by

C n , k ν ( x ) = ( 2 ν ) n , k n ! F k − n k , 2 ν + n k ; ν + k 2 ; 1 − x 2 k .

The first four k -Gegenbauer polynomials are listed as follows:

C o , k ν ( x ) = 1 , C 1 , k ν ( x ) = 2 ν x , C 2 , k ν ( x ) = − ν k + 2 ν ( ν + k ) x 2 , C 3 , k ν ( x ) = − 2 ν k ( ν + k ) x + 4 3 ν ( ν + k ) ( ν + 2 k ) x 3 .

Similar properties in Section 2 for k -Gegenbauer polynomials are obtained using relation (31). Therefore, the following theorem is given without proof:

Theorem 3.2

The k-Gegenbauer polynomials have the following properties:

Explicit formula:
C n , k ν ( x ) = ∑ r = 0 n ( 2 ν ) n , k ν + k 2 n , k k n x − 1 2 k r x + 1 2 k n − r ν + k 2 r , k ν + k 2 n − r , k r ! ( n − r ) ! .
Generating function relation:
∑ n = 0 ∞ C n , k ν ( x ) t n = ( 1 − 2 k x t + k 2 t 2 ) − ν k .
Rodrigues formula:
C n , k ν ( x ) = ( 2 ν ) n , k k n ( x 2 − 1 ) 1 2 − ν k ν + k 2 n , k 2 n n ! D n ( x 2 − 1 ) n + ν k − 1 2 .
Orthogonality:
∫ − 1 1 ( 1 − x 2 ) ν k − 1 2 C n , k ν ( x ) C m , k ν ( x ) d x = 0 , m ≠ n .
Norm value:
‖ C n , k ν ‖ 2 = ∫ − 1 1 ( 1 − x 2 ) ν k − 1 2 [ C n , k ν ( x ) ] 2 d x = k n + 1 ( 2 ν ) n , k π k Γ k ( ν + k 2 ) n ! ( ν + n k ) Γ k ( ν ) .

Theorem 3.3

The k-Gegenbauer polynomials have the following explicit formula:

(34) C n , k ν ( x ) = ∑ r = 0 n 2 ( 2 ν ) n , k x n − 2 r ( x 2 − 1 ) r k r ν + k 2 r , k r ! ( n − 2 r ) ! 2 2 r .

Proof

Let T be denoted as the first member of the generating function relation

(35) ∑ n = 0 ∞ C n , k ν ( x ) t n = ( 1 − 2 k x t + k 2 t 2 ) − ν k .

Then, we have

T = [ 1 − 2 k x t + k 2 t 2 ] − ν k = [ 1 − k x t ] − 2 ν k 1 − k k t 2 ( x 2 − 1 ) ( 1 − k x t ) 2 − ν k = ∑ r = 0 ∞ ( ν ) r , k t 2 r k r ( x 2 − 1 ) r r ! ( 1 − k x t ) 2 ν k + 2 r = ∑ n = 0 ∞ ∑ r = 0 ∞ ( ν ) r , k ( 2 ν + 2 r k ) n , k ( x t ) n k r t 2 r ( x 2 − 1 ) r r ! n ! .

Using the relations

( 2 ν ) 2 r , k ( 2 ν + 2 r k ) n , k = ( 2 ν ) n + 2 r , k

and

( 2 ν ) 2 r , k = 2 2 r ( ν ) r , k ν + k 2 r , k

consecutively, we obtain

T = ∑ n = 0 ∞ ∑ r = 0 ∞ ( 2 ν ) n + 2 r , k x n k r ( x 2 − 1 ) r t n + 2 r ν + k 2 r , k 2 2 r r ! n ! .

By replacing n with n − 2 r , we have

T = ∑ n = 0 ∞ ∑ r = 0 n 2 ( 2 ν ) n , k x n − 2 r k r ( x 2 − 1 ) r t n ν + k 2 r , k 2 2 r r ! ( n − 2 r ) ! .

If the coefficients of t n are equalized, the desired result is obtained.□

Theorem 3.4

The k-Gegenbauer polynomials have the following generating relation:

(36) ∑ n = 0 ∞ ( γ ) n , k ( 2 ν ) n , k C n , k ν ( x ) t n = ( 1 − k x t ) − γ k F k γ 2 , γ 2 + k 2 ; k t 2 ( x 2 − 1 ) ( 1 − k x t ) 2 ν + k 2 ; .

Proof

If C n , k ν ( x ) is replaced by its value in (34) on the left side of equation (36) and then n is replaced by n + 2 r , we obtain

∑ n = 0 ∞ ( γ ) n , k ( 2 ν ) n , k C n , k ν ( x ) t n = ∑ n = 0 ∞ ∑ r = 0 n 2 ( γ ) n , k k r x n − 2 r ( x 2 − 1 ) r t n ν + k 2 r , k r ! ( n − 2 r ) ! 2 2 r = ∑ n = 0 ∞ ∑ r = 0 ∞ ( γ ) n + 2 r , k k r x n ( x 2 − 1 ) r t n + 2 r ν + k 2 r , k r ! n ! 2 2 r .

Here, if some properties of the k -Pochhammer symbol, k -binomial expansion, and definition of k -hypergeometric function are used consecutively, the result of the theorem is obtained.□

Theorem 3.5

The k-Gegenbauer polynomials have the following generating relation:

∑ n = 0 ∞ C n , k ν ( x ) ( 2 ν ) n , k t n = e x t F 1 , k 0 − ; k t 2 ( x 2 − 1 ) 4 ν + k 2 ; .

Proof

It is acquired by techniques similar to those in the proof of Theorem 3.4.□

Theorem 3.6

The following recurrence relations hold true for the k-Gegenbauer polynomials:

n C n , k ν ( x ) = x D C n , k ν ( x ) − k D C n − 1 , k ν ( x ) , n ≥ 1 ,
2 ν C n − 1 , k ν ( x ) = D C n , k ν ( x ) − 2 k x D C n − 1 , k ν ( x ) + k 2 D C n − 2 , k ν ( x ) , n ≥ 2 ,
( n + 1 ) C n + 1 , k ν ( x ) = 2 x ( ν + k n ) C n , k ν ( x ) − k ( 2 ν + k n − k ) C n − 1 , k ν ( x ) , n ≥ 1 ,

where D = d d x .

Proof

(i) If the derivative of both sides of (35) with respect to x and t , respectively, we obtain

∑ n = 0 ∞ D C n , k ν ( x ) t n = 2 ν t ( 1 − 2 k x t + k 2 t 2 ) − ν k − 1 .

and

∑ n = 1 ∞ n C n , k ν ( x ) t n − 1 = 2 ν ( x − k t ) ( 1 − 2 x t + k 2 t 2 ) − ν k − 1 .

If the last two equations are divided side by side and multiplied inside and outside, we have

( x − k t ) ∑ n = 0 ∞ D C n , k ν ( x ) t n = t ∑ n = 1 ∞ n C n , k ν ( x ) t n − 1 .

After making the necessary adjustments, taking into account that C 0 , k ν ( x ) = 1 , we obtain

∑ n = 1 ∞ [ x D C n , k ν ( x ) − k D C n − 1 , k ν ( x ) ] t n = ∑ n = 1 ∞ [ n C n , k ν ( x ) ] t n ,

which, upon equating the coefficients of t n on both sides, yields the desired result.

(ii) Differentiating both sides of (35) with respect to x and applying a similar method in the proof of (i), the desired result easily follows from some simple calculations.

(iii) Differentiating both sides of (35) with respect to t and applying a similar method in the proof of (i), the desired result easily follows from some simple calculations.□

4 k -Legendre polynomials

It is known that if the parameters α and β are replaced by zero in the Jacobi polynomials P n ( α , β ) ( x ) , Legendre polynomials are deduced. Analogously, k -Legendre differential equation and k -Legendre polynomials are defined below and some of their properties are examined. Also, if ν = 1 ⁄ 2 is taken in the Gegenbauer polynomials C n ν ( x ) , then immediately Legendre polynomials are obtained.

Definition 4.1

We introduce the k -Legendre differential equation as follows:

(37) k ( 1 − x 2 ) y ″ − ( k + 1 ) x y ′ + n ( n k + 1 ) y = 0 ,

where k > 0 .

If α = β = 0 in the solution of the k -Jacobi differential equation given by (11), then the following general solution of the k -Legendre differential equation is found.

Theorem 4.1

The general solution of the k-Legendre differential equation given by (37) is

(38) y ( x ) = A F k − n k , n k + 1 ; k + 1 2 ; 1 − x 2 k + B 1 − x 2 k k − 1 2 k F k − 2 n k + k − 1 2 ; 2 n k + k + 1 2 ; 3 k − 1 2 ; 1 − x 2 k .

In view of the definition of the hypergeometric function in the first part of the solution (38), ( − n k ) r , k = 0 for r > n . Thus, the hypergeometric function turns into a hypergeometric polynomial and by selecting A = ( 1 ) n , k n ! , we can define the hypergeometric form of n th order k -Legendre polynomials. This is also found in equality (15), since both α = β = 0 .

Definition 4.2

The k -Legendre polynomials P n , k ( x ) are defined by

P n , k ( x ) = ( 1 ) n , k n ! F k − n k , n k + 1 ; k + 1 2 ; 1 − x 2 k .

The first four k -Legendre polynomials are listed as follows:

P o , k ( x ) = 1 , P 1 , k ( x ) = x , P 2 , k ( x ) = − k 2 + k + 1 2 x 2 , P 3 , k ( x ) = − k k + 1 2 x + 2 3 k + 1 2 2 k + 1 2 x 3 .

If α = β = 0 are taken in equations (17), (22), (25), (26), (27), and (28), respectively, we have the following results for the k -Legendre polynomials.

Theorem 4.2

The k-Legendre polynomials have the following properties:

Explicit formula:
P n , k ( x ) = ∑ r = 0 n ( 1 ) n , k k + 1 2 n , k k n x − 1 2 k r x + 1 2 k n − r k + 1 2 r , k k + 1 2 n − r , k ( n − r ) ! r ! .
Generating function relation:
∑ n = 0 ∞ P n , k ( x ) t n = [ 1 − 2 k x t + k 2 t 2 ] − 1 2 k .
The Bateman-type generating function relation:
∑ n = 0 ∞ P n , k ( x ) t n ( 1 ) n , k k + 1 2 n , k = 0 F 1 , k − ; ( x + 1 ) t 2 k + 1 2 ; F 1 , k 0 − ; ( x − 1 ) t 2 k + 1 2 ; .
Rodrigues formula:
P n , k ( x ) = ( 1 ) n , k k n ( x 2 − 1 ) k − 1 2 k k + 1 2 n , k 2 n n ! D n ( x 2 − 1 ) n + 1 − k 2 k .
Orthogonality:
∫ − 1 1 ( 1 − x 2 ) 1 − k 2 k P n , k ( x ) P m , k ( x ) d x = 0 , m ≠ n .
Norm value:
‖ P n , k ‖ 2 = ∫ − 1 1 ( 1 − x 2 ) 1 − k 2 k [ P n , k ( x ) ] 2 d x = 2 1 k k n + 1 ( 1 ) n , k Γ k k + 1 2 2 n ! ( 1 + 2 n k ) Γ k ( 1 ) .

If ν = 1 2 is taken in Theorem 3.6, we have the next result.

Theorem 4.3

The k-Legendre polynomials have the following recurrence relations:

n P n , k ( x ) = x P n , k ′ ( x ) − k P n − 1 , k ′ ( x ) , n ≥ 1 ,
P n − 1 , k ( x ) = P n , k ′ ( x ) − 2 k x P n − 1 , k ′ ( x ) + k 2 P n − 2 , k ′ ( x ) , n ≥ 2 ,
( n + 1 ) P n + 1 , k ( x ) = ( 2 k n + 1 ) x P n , k ( x ) − [ k 2 ( n − 1 ) + k ] P n − 1 , k ( x ) , n ≥ 1 .

5 Bilinear and bilateral generating functions

In this section, a theorem including several families of bilinear and bilateral generating functions are proven for the k -Jacobi polynomials, similar to the works in [30,31]. Some examples are presented for the theorem. Finally, the corollaries about bilinear and bilateral generating function relations are discussed for k -Gegenbauer and k -Legendre polynomials.

Throughout this section, let n , p ∈ N ; μ , ψ ∈ C ; a s ∈ C \ { 0 } ( s ∈ N 0 ) . Also, let

Ω μ : C r → C \ { 0 }

be a bounded function. Then, the next result is as follows.

Theorem 5.1

Let

Λ μ , ψ [ y 1 , … , y r ; η ] ≔ ∑ s = 0 ∞ a s Ω μ + ψ s ( y 1 , … , y r ) η s

and

θ n , p μ , ψ ( x ; y 1 , … , y r ; z ) ≔ ∑ s = 0 [ n ⁄ p ] a s ( 1 + α + β ) n − p s , k ( 1 + α ) n − p s , k P n − p s , k ( α , β ) ( x ) Ω μ + ψ s ( y 1 , … , y r ) z s .

Then, we have

(39) ∑ n = 0 ∞ θ n , p μ , ψ ( x ; y 1 , … , y r ; η t p ) t n = ( 1 − k t ) − 1 + α + β k F k 1 + α + β 2 , 1 + α + β + k 2 ; 2 t ( x − 1 ) ( 1 − k t ) 2 2 α + k + 1 2 ; Λ μ , ψ [ y 1 , … , y r ; η ] .

Proof

Let T denote the first member of assertion (39). Using θ n , p μ , ψ , we have

T = ∑ n = 0 ∞ ∑ s = 0 [ n ⁄ p ] a s ( 1 + α + β ) n − p s , k ( 1 + α ) n − p s , k P n − p s , k ( α , β ) ( x ) Ω μ + ψ s ( y 1 , … , y r ) η s t n − p s

Replacing n by n + p s , we obtain

T = ∑ n = 0 ∞ ∑ s = 0 ∞ a s ( 1 + α + β ) n , k ( 1 + α ) n , k P n , k ( α , β ) ( x ) Ω μ + ψ s ( y 1 , … , y r ) η s t n

= ∑ n = 0 ∞ ( 1 + α + β ) n , k ( 1 + α ) n , k P n , k ( α , β ) ( x ) t n ∑ s = 0 ∞ a s Ω μ + ψ s ( y 1 , … , y r ) η s = ( 1 − k t ) − 1 + α + β k F k 1 + α + β 2 , 1 + α + β + k 2 ; 2 t ( x − 1 ) ( 1 − k t ) 2 2 α + k + 1 2 ; Λ μ , ψ [ y 1 , … , y r ; η ] ,

which is the right member of (39).□

It is possible to give many applications of Theorem 5.1 by making appropriate choices of the multivariable functions Ω μ + ψ s ( y 1 , … , y r ) . Since this multivariable function is very general, we may deduce a number of particular formulas from this result. Now, we present the following two examples.

Example 5.1

The Lagrange polynomials g n ( γ , λ ) ( y , z ) are generated by (see [28])

(40) ( 1 − y t ) − γ ( 1 − z t ) − λ = ∑ n = 0 ∞ g n ( γ , λ ) ( y , z ) t n ,

where ∣ t ∣ < min { ∣ y ∣ − 1 , ∣ z ∣ − 1 } . If we take r = 2 , a s = 1 , μ = 0 , ψ = 1 and replace the function Ω μ + ψ s in Theorem 5.1 with the Lagrange polynomials, then, using the relation (40) and Theorem 5.1, we obtain

∑ n = 0 ∞ ∑ s = 0 [ n ⁄ p ] ( 1 + α + β ) n − p s , k ( 1 + α ) n − p s , k P n − p s , k ( α , β ) ( x ) g s ( γ , λ ) ( y , z ) η s t n − p s = ( 1 − y η ) − γ ( 1 − z η ) − λ ( 1 − k t ) − 1 + α + β k F k 1 + α + β 2 , 1 + α + β + k 2 ; 2 t ( x − 1 ) ( 1 − k t ) 2 2 α + k + 1 2 ; ,

which is a class of bilateral generating functions for the k -Jacobi polynomials and the Lagrange polynomials.

Example 5.2

Taking r = 1 , a s = ( 1 + α + β ) s , k ( 1 + α ) s , k , μ = 0 , ψ = 1 and taking the k -Jacobi polynomials instead of the function Ω μ + ψ s in Theorem 5.1 and also using (22), we obtain the following class of bilinear generating functions for the k -Jacobi polynomials:

∑ n = 0 ∞ ∑ s = 0 [ n ⁄ p ] ( 1 + α + β ) s , k ( 1 + α + β ) n − p s , k ( 1 + α ) s , k ( 1 + α ) n − p s , k P n − p s , k ( α , β ) ( x ) P s , k ( α , β ) ( x ) η s t n − p s = [ ( 1 − k t ) ( 1 − k η ) ] − 1 + α + β k F k 1 + α + β 2 , 1 + α + β + k 2 ; 2 t ( x − k ) ( 1 − k t ) 2 2 α + k + 1 2 ; × F k 1 + α + β 2 , 1 + α + β + k 2 ; 2 η ( x − k ) ( 1 − k η ) 2 2 α + k + 1 2 ; .

Corollary 5.1

If we take α = β = ν − 1 2 and use relation (31) in Theorem 5.1, we obtain the result containing families of bilinear and bilateral generating function relations for the k-Gegenbauer polynomials.

Corollary 5.2

If we take α = β = 0 in Theorem 5.1, we obtain the result containing families of bilinear and bilateral generating function relations for the k-Legendre polynomials.

6 Conclusion

In recent years, many mathematicians have defined various generalizations of Jacobi polynomials and studied their applications. k -generalization of functions is one of the new techniques.

In this study, various features of the k -Jacobi polynomials, the k -Gegenbauer polynomials, and the k -Legendre polynomials which are generalization of the Jacobi, Gegenbauer, and Legendre polynomials are introduced and demonstrated. For k = 1 , k -version of these polynomials reduces to the known Jacobi, Gegenbauer, and Legendre polynomials, respectively. Bilinear and bilateral generating function relations are also studied and some examples are displayed.

After definitions of found, new studies can be done on these polynomials. For example, two variable version of the k -Jacobi polynomials can be investigated using the technique in [29]. Integral representations, multilateral and multilinear generating function relations, and linearization coefficients for these new polynomials can also be derived.

For all terminology below, readers are referred to [27].

Using the following transitions and taking into account the definitions of k -Jacobi and k -Gegenbauer polynomials given in this article, one can define k -generalizations of the following polynomials.

The special case β = α of the Jacobi polynomials is called the ultraspherical polynomials and is denoted P n ( α , α ) ( x ) .
The Tchebycheff polynomials T n ( x ) and U n ( x ) of the first and second kinds, respectively, are special ultraspherical polynomials. The various properties of these polynomials can easily be derived from those of Jacobi polynomials by setting α = β = − 1 2 and α = β = 1 2 , and using the relations
T n ( x ) = n ! 1 2 n P n ( − 1 \ 2 , − 1 \ 2 ) ( x )
and
U n ( x ) = ( n + 1 ) ! 3 2 n P n ( 1 \ 2,1 \ 2 ) ( x ) .
The Laguerre polynomials are limiting cases of the Jacobi polynomials, and we have the relationship
L n ( α ) ( x ) = lim ∣ β ∣ → ∞ P n ( α , β ) 1 − 2 x β .
Other limiting relationship between Hermite and Gegenbauer polynomials:
H n ( x ) = n ! lim ∣ ν ∣ → ∞ ν − n \ 2 C n ν x ν .
The two-parameter Bessel polynomials provided by
y n ( x , α , β ) = lim ∣ γ ∣ → ∞ n ! ( γ ) n P n ( γ − 1 , α − γ − 1 ) 1 + 2 γ x β

are another member of the family of classical orthogonal polynomials, which is a limit case of the Jacobi polynomials.

Acknowledgments

The author would like to thank the referees for valuable comments that improved the article.

Funding information: The author claims there was no funding involved.
Author contribution: The author confirms sole responsibility for the design of the study, the presentation of the results, and the preparation of the manuscript.
Conflict of interest: The author declares no conflict of interest.
Data availability statement: Not applicable.

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Received: 2024-12-14

Revised: 2025-04-15

Accepted: 2025-05-27

Published Online: 2025-10-07

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0157

Keywords for this article

Jacobi polynomials; Legendre polynomials; hypergeometric function; Rodrigues formula; orthogonal polynomials; Pochhammer symbol

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