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Article

Explicit Properties of Apostol-Type Frobenius–Euler Polynomials Involving q-Trigonometric Functions with Applications in Computer Modeling

1
Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China
2
Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia
3
Department of Basic Sciences, Faculty of Engineering, Hasan Kalyoncu University, Gaziantep TR-27010, Turkey
4
Department of Mathematics, Hannam University, Daejeon 34430, Republic of Korea
*
Authors to whom correspondence should be addressed.
Mathematics 2023, 11(10), 2386; https://doi.org/10.3390/math11102386
Submission received: 4 May 2023 / Revised: 17 May 2023 / Accepted: 18 May 2023 / Published: 20 May 2023

Abstract

:
In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results.

1. Introduction

Recently, many authors have considered and applied the generating functions techniques to new families of special polynomials, including two parametric kinds of polynomials, such as Bernoulli, Euler, Genocchi, etc. (see [1,2,3,4,5,6,7,8,9,10]). They have firstly derived the basic identities of these polynomials. Additionally, they have established more identities and relations among trigonometric functions, using two parametric kinds of polynomials by using generating functions. By applying the partial derivative operator to these generating functions, derivative formulae, and finite combinatorial sums involving the special polynomials and numbers are obtained. We would like to note that these special polynomials facilitate the derivation of various helpful properties in a fairly straightforward way and lead to introducing new families of special polynomials. The Apostol-type polynomials appear in combinatorial mathematics and play an important role in theory, generalization, applications and modeling; thus, many number theorists and combinatorics experts have extensively investigated their properties and obtained a series of interesting results (see [5,8,9,11,12,13]). Inspired by the above polynomials, in this study, we are in a position to state the parametric kinds of Apostol-type Frobenius–type Euler polynomials by introducing the two specific q-analogues of exponential generating functions. Additionally, we prove many formulas and relations for these polynomials, including some implicit summation formulas, differentiation rules and correlations with the earlier polynomials by utilizing some series manipulation methods. Additionally, as an application, we show the zero values of q-Apostol-type Frobenius–type Euler polynomials using tables and draw some graphical representations.
We begin by stating the following definitions and notations of q-calculus reviewed here, which are taken from (see [14]):
A q-analogue of the shifted factorial ( a ) n is given by
( a ; q ) 0 = 1 , ( a ; q ) n = m = 0 n 1 ( 1 q m a ) , n N .
A q-analogue of a complex number a and of the factorial function are given by
[ a ] q = 1 q a 1 q , ( q C \ { 1 } ; a C ) ,
[ n ] q ! = m = 1 n [ m ] q = [ 1 ] q [ 2 ] q [ n ] q = ( q ; q ) n ( 1 q ) n , q 1 ; n N ,
[ 0 ] q ! = 1 , q C ; 0 < q < 1 .
The Gauss q-binomial coefficient n k q is given by
n k q = [ n ] q ! [ k ] q ! [ n k ] q ! = ( q ; q ) n ( q ; q ) k ( q ; q ) n k , k = 0 , 1 , , n .
The q-analogue of the function ( x + y ) q n is given by
( x + y ) q n = k = 0 n n k q q k ( k 1 ) / 2 x n k y k , n N 0 .
The q-analogues of exponential functions are given by
e q ( x ) = n = 0 x n [ n ] q ! = 1 ( ( 1 q ) x ; q ) , 0 < q < 1 ; x < 1 q 1 ,
E q ( x ) = n = 0 q ( n 2 ) [ n ] q ! x n = ( ( 1 q ) x ; q ) , 0 < q < 1 ; x C .
These two functions are related by the equation (see [14])
e q ( x ) E q ( x ) = 1 .
A q-derivative operator of a function is defined by
D q , z f ( z ) : = D q f ( z ) = f ( q z ) f ( z ) q z z , 0 < q < 1 ,
and D q f ( 0 ) = f ( 0 ) provided that f is differentiable at x = 0 .
A q-derivative fulfills the following product and quotient rules
D q , z ( f ( z ) g ( z ) ) = f ( z ) D q , z g ( z ) + g ( q z ) D q , z f ( z ) ,
D q , z f ( z ) g ( z ) = g ( q z ) D q , z f ( z ) f ( q z ) D q , z g ( z ) g ( z ) g ( q z ) .
The Apostol-type q-Bernoulli polynomials B n , q ( α ) ( x ; λ ) of order α , the Apostol-type q-Euler polynomials E n , q ( α ) ( x ; λ ) of order α and the Apostol-type q-Genocchi polynomials G n , q ( α ) ( x ; λ ) of order α are defined by (see [15,16]):
t λ e q ( t ) 1 α e x t = n = 0 B n , q ( α ) ( x ; λ ) t n [ n ] q ! ,
2 λ e q ( t ) + 1 α e x t = n = 0 E n , q ( α ) ( x ; λ ) t n [ n ] q ! ,
2 t λ e q ( t ) + 1 α e x t = n = 0 G n , q ( α ) ( x ; λ ) t n [ n ] q ! ,
respectively.
Clearly, we can obtain
B n , q ( α ) ( λ ) = B n , q ( α ) ( 0 ; λ ) , E n , q ( α ) ( λ ) = E n , q ( α ) ( 0 ; λ ) , G n , q ( α ) ( λ ) = G n , q ( α ) ( 0 ; λ ) ,
and
B n , q ( 1 ) ( x ; λ ) = B n , q ( x ; λ ) , E n , q ( 1 ) ( x ; λ ) = E n , q ( x ; λ ) , G n , q ( 1 ) ( x ; λ ) = G n , q ( x ; λ ) .
Let u C with u 1 and ξ R . The Apostol-type q-Frobenius–Euler polynomials H n , q ( α ) ( x , y ; u ; λ ) of order α C are defined by (see [11,12]):
1 u λ e q ( t ) u α e q ( x t ) E q ( y t ) = n = 0 H n , q ( α ) ( x , y ; u ; λ ) t n [ n ] q ! , z < ln λ u .
It is obvious that
H n , q ( α ) ( u ; λ ) = H n , q ( α ) ( 0 , 0 ; u ; λ ) lim q 1 H n , q ( α ) ( x , y ; u ; λ ) = H n ( α ) ( x + y ; u ; λ ) .
Kang et al. [2,4] introduced the q-Bernoulli and q-Euler polynomials defined by
z e q ( z ) 1 e q ( ξ z ) C O S q ( η z ) = j = 0 B j , q ( ( ξ + i η ) q ) + B j ( ( ξ i η ) q ) 2 z j [ j ] q ! = j = 0 B j , q ( C ) ( ξ , η ) z j [ j ] q ! ,
z e q ( z ) 1 e q ( ξ z ) S I N q ( η z ) = j = 0 B j , q ( ( ξ + i η ) q ) B j , q ( ( ξ i η ) q ) 2 i z j [ j ] q ! = j = 0 B j , q ( S ) ( ξ , η ) z j [ j ] q ! ,
and
2 e q ( z ) + 1 e q ( ξ z ) C O S q ( η z ) = j = 0 E j , q ( ( ξ + i η ) q ) + E j , q ( ( ξ i η ) q ) 2 z j [ j ] q ! = j = 0 E j , q ( C ) ( ξ , η ) z j [ j ] q ! ,
2 e q ( z ) + 1 e q ( ξ z ) S I N q ( η z ) = j = 0 E j ( ( ξ + i η ) q ) E j ( ( ξ i η ) ) q 2 i z j [ j ] q ! = j = 0 E j , q ( S ) ( ξ , η ) z j [ j ] q ! ,
respectively.
Additionally, they have proved that (see [2,4]):
e q ( ξ z ) C O S q ( η z ) = r = 0 C r , q ( ξ , η ) z r [ r ] q ! ,
and
e q ( ξ z ) S I N q ( η z ) = r = 0 S r , q ( ξ , η ) z r [ r ] q ! ,
where
C r , q ( ξ , η ) = j = 0 [ r 2 ] r 2 j q ( 1 ) j q 2 j 1 ξ r 2 j η 2 j ,
and
S r , q ( ξ , η ) = j = 0 [ r 1 2 ] r 2 j + 1 q ( 1 ) j q ( 2 j + 1 ) j ξ r 2 j 1 η 2 j + 1 .

2. q-Apostol-Type Frobenius–Euler Polynomials of Complex Variable

In this section, we consider the q-Cosine and q-Sine Apostol-type Frobenius–Euler polynomials of a complex variable and deduce some identities of these polynomials. First, we present the following definition.
1 u λ e q ( t ) u α e q ( x t ) E q ( i t y ) = n = 0 H n , q ( α ) ( ( ξ + i y ) q ; u ; λ ) t n [ n ] q ! .
It is well-known from ([4] Definition 5) that
e q ( x t ) E q ( i t y ) = e q ( x t ) ( C O S q ( y t ) + i S I N q ( y t ) ) .
Thus, by (18) and (19), we have
n = 0 H n , q ( α ) ( ( x + i y ) q ; u ; λ ) t n n ! = 1 u λ e q ( t ) u α e q ( x t ) E q ( i t y )
= 1 u λ e q ( t ) u α e q ( x t ) ( C O S q ( y z ) + i S I N q ( y z ) ) ,
and
n = 0 H n , q ( α ) ( ( x i y ) q ; u ; λ ) t n [ n ] q ! = 1 u λ e q ( t ) u α e q ( x t ) E q ( i t y )
= 1 u λ e q ( t ) u α e q ( x t ) ( C O S q ( y t ) i S I N q ( y t ) ) .
From (20) and (21), we get
1 u λ e q ( t ) u α e q ( x t ) C O S q ( y t ) = n = 0 H n , q ( α ) ( ( x + i y ) q ; u ; λ ) + H n , q ( α ) ( ( x i y ) q ; u ; λ ) 2 t n [ n ] q ! ,
and
1 u λ e q ( t ) u α e q ( x t ) S I N q ( y t ) = n = 0 H n , q ( α ) ( ( x + i y ) q ; u ; λ ) H n , q ( α ) ( ( x i y ) q ; u ; λ ) 2 t n [ n ] q ! .
Definition 1.
Let j 0 . We define two parametric kinds of q-Cosine Apostol-type Frobenius–Euler polynomials H n , q ( α , c ) ( x , y ; u ; λ ) and q-Sine Apostol-type Frobenius–Euler polynomials H n , q ( α , s ) ( x , y ; u ; λ ) , for a non negative integer n, by
1 u λ e q ( t ) u α e q ( x t ) C O S q ( y t ) = n = 0 H j , q ( α , c ) ( x , y ; u ; λ ) t n [ n ] q ! ,
and
1 u λ e q ( t ) u α e q ( x t ) S I N q ( y t ) = n = 0 H n , q ( α , s ) ( x , y ; u ; λ ) t n [ n ] q ! ,
respectively.
Note that H n , q ( α , c ) ( 0 , 0 ; u ; λ ) = H n , q ( u ; λ ) , H n , q ( α , s ) ( 0 , 0 ; u ; λ ) = 0 , ( n 0 ) .
From (22)–(25), we have
H n , q ( α , c ) ( x , y ; u ; λ ) = H n , q ( ( x + i y ) q ; u ; λ ) + H n , q ( ( x i y ) q ; u ; λ ) 2 ,
H n , q ( α , s ) ( x , y ; u ; λ ) = H n , q ( ( x + i y ) q ; u ; λ ) H n , q ( ( x i y ) q ; u ; λ ) 2 i .
Remark 1.
For x = 0 in (24) and (25), we obtain
1 u λ e t u α C O S q ( y t ) = j = 0 H n , q ( α , c ) ( y ; u ; λ ) t n [ n ] q ! ,
and
1 u λ e t u α S I N q ( y t ) = n = 0 H n , q ( α , s ) ( y ; u ; λ ) t n [ n ] q ! ,
respectively.
It is clear that
H n , q ( α , c ) ( 0 ; u ; λ ) = H n , q ( u ; λ ) , H n , q ( α , s ) ( 0 ; u ; λ ) = 0 , ( n 0 ) .
Now, we provide some basic properties of these polynomials.
Theorem 1.
Let n 0 . Then,
H n , q ( α , c ) ( y ; u ; λ ) = v = 0 [ n 2 ] n + v 2 v q ( 1 ) v q ( 2 v 1 ) v y 2 v H n 2 v , q ( u ; λ ) ,
and
H j , q ( α , s ) ( u ; λ ) = v = 0 [ n 1 2 ] n + v 2 v + 1 q ( 1 ) v q ( 2 v + 1 ) v y 2 v + 1 H n 2 v 1 , q ( u ; λ ) .
Proof. 
By (28) and (29), we can derive the following equations
n = 0 H n , q ( α , c ) ( y ; u ; λ ) t n [ n ] q ! = 1 u λ e q ( t ) u α C O S q ( y t )
= n = 0 H n , q ( α ) ( u ; λ ) t n [ n ] q ! v = 0 ( 1 ) v q ( 2 v 1 ) v y 2 v t v [ 2 v ] q ! .
= n = 0 v = 0 [ n 2 ] n + v 2 v q ( 1 ) v q ( 2 v 1 ) v y 2 v H n 2 v , q ( u ; λ ) t n [ n ] q ! ,
and
n = 0 H n , q ( α , s ) ( y ; u ; λ ) t n n ! = 1 u λ e q ( t ) u α S I N q ( y t )
= n = 0 v = 0 [ n 1 2 ] n 2 v + 1 q ( 1 ) v q ( 2 v + 1 ) v y 2 v + 1 H j 2 v 1 , q ( u ; λ ) t n [ n ] q ! .
Therefore, with (32) and (33), we get (30) and (31). □
Theorem 2.
Let n 0 . Then,
H n , q ( α ) ( ( x + i y ) q ; u ; λ ) = k = 0 n n k q ( x + i y ) q k H n k , q ( α ) ( u ; λ )
= k = 0 n n k q ( i y ) k H j k , q ( α ) ( x ; u ; λ ) ,
and
H n , q ( α ) ( ( x i y ) q ; u ; λ ) = k 0 n n k q ( x i y ) q k H n k , q ( α ) ( u ; λ )
= k = 0 n n k q ( 1 ) k ( i y ) k H n k , q ( x ; u ; λ ) .
Proof. 
By using (20) and (21), we obtain (34) and (35). So, we omit the proof. □
Theorem 3.
Let n 0 . Then,
H n , q ( α , c ) ( x , y ; u ; λ ) = k = 0 n n k q H k , q ( α ) ( u ; λ ) C n k , q ( x , y ) ,
and
H n , q ( α , s ) ( x , y ; u ; λ ) = k = 0 n n k q H k , q ( u ; λ ) S n k , q ( x , y ) .
Proof. 
Consider
n = 0 a n t n n ! k = 0 b k t k k ! = n = 0 k = 0 j a n k b k t n n ! .
Now,
n = 0 H j , q ( α , c ) ( x , y ; u ; λ ) t n [ n ] q ! = 1 u λ e q ( t ) u α e q ( x t ) C O S q ( y t )
= k = 0 H k , q ( α ) ( u ; λ ) t k k ! n = 0 C n , q ( x , y ) t n [ n ] q !
= n = 0 k = 0 n n k q H k , q ( α ) ( u ; λ ) C n k , q ( x , y ) t n [ n ] q ! ,
which proves (36). The proof of (37) is similar.
By using Definition 1, we can easily obtain the following Theorems. So, we omit the proofs.
Theorem 4.
Let j 0 . Then,
H j , q ( α , c ) ( x + s , y ; u ; λ ) = k = 0 n n k q H k , q ( α , c ) ( x , y ; u ; λ ) r n k ,
and
H n , q ( α , s ) ( x , y ; u ; λ ) = k = 0 n n k q H k , q ( α , s ) ( x , y ; u ; λ ) r n k .
Theorem 5.
Let j 1 . Then,
x H n , q ( α , c ) ( x , y ; u ; λ ) = [ n ] q H n 1 , q ( α , c ) ( x , y ; u ; λ ) ,
y H n , q ( α , c ) ( x , y ; u ; λ ) = [ n ] q H n 1 , q ( α , s ) ( x , q y ; u ; λ ) ,
and
x H n , q ( α , s ) ( x , y ; u ; λ ) = [ n ] q H n 1 , q ( α , s ) ( x , y ; u ; λ ) ,
y H n , q ( α , s ) ( x , y ; u ; λ ) = [ n ] q H n 1 , q ( α , c ) ( x , q y ; u ; λ ) .
Theorem 6.
Let n be a nonnegative integer, the following formulas hold true.
λ H n , q ( α , c ) ( 1 , y ; u ; λ ) u H n , q ( α , c ) ( 0 , y ; u ; λ )
= ( 1 u ) H n , q ( α 1 , c ) ( 0 ; y ; u ; λ ) ,
λ H n , q ( α , s ) ( 1 , y ; u ; λ ) u H n , q ( α , s ) ( 0 , y ; u ; λ )
= ( 1 u ) H n , q ( α 1 , s ) ( 0 ; y ; u ; λ ) ,
Theorem 7.
The following relations hold true.
H n , q ( α + β , c ) ( x , y ; u ; λ ) = m = 0 n n m q H n m , q ( α , c ) ( x , y ; u ; λ ) H m , q ( β ) ( u ; λ ) ,
H n , q ( α + β , s ) ( x , y ; u ; λ ) = m = 0 n n m q H n m , q ( α , s ) ( x , y ; u ; λ ) H m , q ( β ) ( u ; λ ) ,
H n , q ( α β , c ) ( x , y ; u ; λ ) = m = 0 n n m q H n m , q ( α , c ) ( x , y ; u ; λ ) H m , q ( β ) ( u ; λ ) ,
H n , q ( α β , s ) ( x , y ; u ; λ ) = m = 0 n n m q H n m , q ( α , s ) ( x , y ; u ; λ ) H m , q ( β ) ( u ; λ ) ,
Theorem 8.
Let x , y , and r be any real numbers. Then, we have
(i) 
H n , q ( α , c ) ( ( x + r ) q , y ; u ; λ ) + H n , q ( α , s ) ( ( x r ) q , y ; u ; λ )
= k = 0 n n l q q ( n l 2 ) r n l H n , q ( α , c ) ( x , y ; u ; λ ) + ( 1 ) n k H n , q ( α , s ) ( x , y ; u ; λ ) ,
(ii) 
H n , q ( α , s ) ( ( x + r ) q , y ; u ; λ ) + H n , q ( α , c ) ( ( x r ) q , y ; u ; λ )
= k = 0 n n l q q ( n l 2 ) r n l H n , q ( α , s ) ( x , y ; u ; λ ) + ( 1 ) n k H n , q ( α , c ) ( x , y ; u ; λ ) .
Corollary 1.
Let j 0 . Then,
H n , q ( c ) ( ( x + r ) q , y ; u ; λ ) + H n , q ( c ) ( ( x r ) q , y ; u ; λ )
= k = 0 n n k q q ( k 2 ) r k H n k , q ( c ) ( x , y ; u ; λ ) + ( 1 ) k H n k , q ( c ) ( x , y ; u ; λ ) ,
and
H n , q ( s ) ( ( x + r ) q , y ; u ; λ ) + H n , q ( s ) ( ( x r ) q , y ; u ; λ )
= k = 0 n n k q q ( k 2 ) r k H n k , q ( s ) ( x , y ; u ; λ ) + ( 1 ) k H n k , q ( s ) ( x , y ; u ; λ ) .
Corollary 2.
For r = 1 in Theorem 8, we obtain
H n , q ( c ) ( ( x + 1 ) q , y ; u ; λ ) + H n , q ( s ) ( ( x 1 ) q , y ; u ; λ )
= k = 0 n n k q q ( k 2 ) r k H n k , q ( c ) ( x , y ; u ; λ ) + ( 1 ) k H n k , q ( s ) ( x , y ; u ; λ ) ,
and
H n , q ( s ) ( ( x + 1 ) q , y ; u ; λ ) + H n , q ( c ) ( ( x 1 ) q , y ; u ; λ )
= k = 0 n n k q q ( n k 2 ) r n k H k , q ( s ) ( x , y ; u ; λ ) + ( 1 ) n k H k , q ( c ) ( x , y ; u ; λ ) .

3. Summation Formulas for q-Cosine and q-Sine Apostol-Type Frobenius–Euler Polynomials

In this section, we derive some correlations for the q-cosine and q-sine Apostol-type Frobenius–Euler polynomials of order α associated with the q-Bernoulli, Euler, and Genocchi polynomials and the q-Stirling numbers of the second kind. We first provide the following theorems.
Theorem 9.
The following results hold true:
( 2 u 1 ) k = 0 n n k q H k , q ( x , 0 ; u ; λ ) H n k , q ( c ) ( 0 , y ; 1 u ; λ )
= u H n , q ( c ) ( x , y ; u ; λ ) ( 1 u ) H n , q ( c ) ( x , y ; 1 u ; λ ) ,
and
( 2 u 1 ) k = 0 n n k q H k , q ( x , 0 ; u ; λ ) H n k , q ( s ) ( 0 , y ; 1 u ; λ )
= u H n , q ( s ) ( x , y ; u ; λ ) ( 1 u ) H n , q ( s ) ( x , y ; 1 u ; λ ) .
Proof. 
We set
( 2 u 1 ) ( λ e q ( t ) u ) ( λ e q ( t ) ( 1 u ) ) = 1 λ e q ( t ) u 1 λ e q ( t ) ( 1 u ) .
From the above equation, we see that
( 2 u 1 ) ( 1 u ) e q ( x t ) ( 1 ( 1 u ) ) C O S q ( y t ) ( λ e q ( t ) u ) ( λ e q ( t ) ( 1 u ) )
= ( 1 u ) e q ( x t ) u C O S q ( y t ) λ e q ( t ) u ( 1 u ) e q ( x t ) C O S q ( y t ) ( 1 ( 1 u ) ) λ e q ( t ) ( 1 u ) ,
which when using Equations (9) and (24) on both sides, we can obtain
( 2 u 1 ) k = 0 H k , q ( x , 0 ; u ; λ ) t k [ k ] q ! n = 0 H n , q ( c ) ( 0 , y ; 1 u ; λ ) t n [ n ] q !
= u n = 0 H n , q ( c ) ( x , y ; u ; λ ) t n [ n ] q ! ( 1 u ) n = 0 H n , q ( c ) ( x , y ; 1 u ; λ ) t n [ n ] q ! .
By applying the Cauchy product rule in the above equation and then equating the coefficients of like powers of t on both sides of the resultant equation, assertion (56) follows. Similarly, we obtain (57). □
Theorem 10.
The following relations hold true:
u H n , q ( c ) ( x , y ; u ; λ ) = k = 0 n n k q H n , q ( c ) ( x , y ; u ; λ ) ( 1 u ) C n , q ( x , y ) ,
and
u H n , q ( s ) ( x , y ; u ; λ ) = k = 0 n n k q H n , q ( s ) ( x , y ; u ; λ ) ( 1 u ) S n , q ( x , y ) .
Proof. 
Consider the following identity
u λ ( λ e q ( t ) u ) e q ( t ) = 1 ( λ e q ( t ) u ) 1 λ e q ( t ) .
Evaluating the following fraction using the above identity, we find
u ( 1 u ) e q ( x t ) C O S q ( y t ) λ ( λ e q ( t ) u ) e q ( t ) = ( 1 u ) e q ( x t ) C O S q ( y t ) λ e q ( t ) u ( 1 u ) e q ( x t ) C O S q ( y t ) λ e q ( t )
u n = 0 H n , q ( c ) ( x , y ; u ; λ ) t n [ n ] q ! = λ n = 0 H n , q ( c ) ( x , y ; u ; λ ) t n [ n ] q ! k = 0 t k [ k ] q ! ( 1 u ) n = 0 C n , q ( x , y ) t n [ n ] q ! .
By applying the Cauchy product rule in the above equation and then equating the coefficients of like powers of t on both sides of the resultant equation, assertion (58) follows. Similarly, we obtain (59). □
The following Theorems can be easily derived by making use of the definitions of used polynomials and series manipulations. So, we omit the proofs.
Theorem 11.
The following relation holds true:
H n , q ( α , c ) ( x , y ; u ; λ )
= 1 1 u k = 0 n n k q λ H n k , q ( u ; λ ) H k , q ( α , c ) ( x , y ; u ; λ )
u H n k , q ( u ; λ ) H k , q ( α , c ) ( x , y ; u ; λ ) .
and
H n , q ( α , s ) ( x , y ; u ; λ )
= 1 1 u k = 0 n n k q λ H n k , q ( u ; λ ) H k , q ( α , s ) ( x , y ; u ; λ )
u H n k , q ( u ; λ ) H k , q ( α , s ) ( x , y ; u ; λ ) .
Theorem 12.
The following relations hold true:
H n , q ( α , c ) ( x , y ; u ; λ ) = k = 0 n + 1 n + 1 k q λ r = 0 k k r q B k r , q ( x ; λ ) B k , q ( x ; λ )
× H n k + 1 , q ( α , c ) ( 0 , y ; u ; λ ) .
and
H n , q ( α , s ) ( x , y ; u ; λ ) = k = 0 n + 1 n + 1 k q λ r = 0 k k r q B k r , q ( x ; λ ) B k , q ( x ; λ )
× H n k + 1 , q ( α , s ) ( 0 , y ; u ; λ ) .
Theorem 13.
The following relations hold true:
H n , q ( α , c ) ( x , y ; u ; λ ) = 1 2 k = 0 n n k q ( λ r = 0 k k r q E k r , q ( λ ) + E k , q ( λ ) )
× H n k , q ( α , c ) ( x , y ; u ; λ ) .
and
H n , q ( α , s ) ( x , y ; u ; λ ) = 1 2 k = 0 n n k q ( λ r = 0 k k r q E k r , q ( λ ) + E k , q ( λ ) )
× H n k , q ( α , s ) ( x , y ; u ; λ ) .
Theorem 14.
The following relations hold true:
H n , q ( α , c ) ( x , y ; u ; λ ) = 1 2 k = 0 n + 1 n + 1 k q ( λ r = 0 k k r q G k r , q ( λ ) + G k , q ( λ ) )
× H n k + 1 , q ( α , c ) ( x , y ; u ; λ ) .
and
H n , q ( α , s ) ( x , y ; u ; λ ) = 1 2 k = 0 n + 1 n + 1 k q ( λ r = 0 k k r q G k r , q ( λ ) + G k , q ( λ ) )
× H n k + 1 , q ( α , s ) ( x , y ; u ; λ ) .
Theorem 15.
Let α and γ be nonnegative integers. The following relations hold true:
1 u u α C n , q ( x , y ) = α ! l = 0 n n l q H n l , q ( α , c ) ( x , y ; u ; λ ) S ( l , α , λ u : q ) ,
1 u u α S n , q ( x , y ) = α ! l = 0 n n l q H n l , q ( α , s ) ( x , y ; u ; λ ) S ( l , α , λ u : q ) ,
H n , q ( α γ , c ) ( x , y ; u ; λ )
= γ ! u 1 u γ α ! l = 0 n n l q H n l , q ( α , c ) ( x , y ; u ; λ ) S ( l , α , λ u : q ) ,
and
H n , q ( α γ , s ) ( x , y ; u ; λ )
= γ ! u 1 u γ α ! l = 0 n n l q H n l , q ( α , s ) ( x , y ; u ; λ ) S ( l , α , λ u : q ) .
Theorem 16.
The following relations hold true:
H n , q ( α , c ) ( x , y ; u ; λ )
= j = 0 n l = j n α + j 1 j q j ! n l q ( 1 u ) j S ( l , j ; λ : q ) ( λ 1 ) l j C n l , q ( x , y ) .
and
H n , q ( α , s ) ( x , y ; u ; λ )
= j = 0 n l = j n α + j 1 j q j ! n l q ( 1 u ) j S ( l , j ; λ : q ) ( λ 1 ) l j S n l , q ( x , y ) .
Theorem 17.
The following relationships hold true:
H n , q ( α , c ) ( x , y ; u ; λ ) = k = 0 n ( α ) k ( u 1 ) k S n , q ( k , c ) ( x , y ; λ )
and
H n , q ( α , s ) ( x , y ; u ; λ ) = k = 0 n ( α ) k ( u 1 ) k S n , q ( k , s ) ( x , y ; λ ) ,
where
e q ( x t ) C O S q ( y t ) ( λ e q ( t ) 1 ) k [ k ] q ! = n = k S n , q ( k , c ) ( x , y ; λ ) t n [ n ] q !
and
e q ( x t ) S I N q ( y t ) ( λ e q ( t ) 1 ) k [ k ] q ! = n = k S n , q ( k , s ) ( x , y ; λ ) t n [ n ] q ! .

4. Symmetry Identities for q-Cosine and q-Sine Apostol-Type Frobenius–Euler Polynomials

In this section, we describe the general symmetry identities for the q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and generalized Apostol-type Frobenius–Euler polynomials by applying the generating functions (9), (24) and (25). We begin with the following theorem.
Theorem 18.
Let a , b , > 0 with a b and j 0 . Then,
(i) 
k = 0 n n k q b k a n k H n k , q ( α , c ) ( b x , b y ; u ; λ ) H k , q ( α , c ) ( a x , a y ; u ; λ )
= k = 0 n n k q a k b n k H n k , q ( α , c ) ( a x , a y ; u ; λ ) H k , q ( α , c ) ( b x , b y ; u ; λ ) ,
(ii) 
k = 0 n n k q b k a n k H n k , q ( α , s ) ( b x , b y ; u ; λ ) H k , q ( α , s ) ( a x , a y ; u ; λ )
= k = 0 n n k q a k b n k H n k , q ( α , s ) ( a x , a y ; u ; λ ) H k , q ( α , s ) ( b x , b y ; u ; λ ) .
Proof. 
Let
A ( z ) = ( 1 u ) 2 ( e q ( a b x t ) C O S q ( a b y t ) ) 2 ( λ e q ( a z ) u ) ( λ e q ( b z ) u ) 2 α .
Then, the expression for A ( z ) is symmetric in a and b, and we obtain
A ( z ) = n = 0 H n , q ( α , c ) ( b x , b y ; u ; λ ) ( a t ) n [ n ] q ! k = 0 H k , q ( α , c ) ( a x , a y ; u ; λ ) ( b t ) k [ k ] q !
= n = 0 ( k = 0 n j k q b k a j k H n k , q ( α , c ) ( b x , b y ; u ; λ ) H k , q ( α , c ) ( a x , a y ; u ; λ ) ) t n [ n ] q ! .
Similarly, we can show that
A ( z ) = n = 0 H n , q ( α , c ) ( a x , a y ; u ; λ ) ( b t ) n [ n ] q ! k = 0 H k , q ( α , c ) ( b x , b y ; u ; λ ) ( a t ) k k !
= n = 0 ( k = 0 n n k q a k b n k H n k , q ( α , c ) ( a x , a y ; u ; λ ) H k , q ( α , c ) ( b x , b y ; u ; λ ) ) t n [ n ] q ! .
On comparing the coefficients of t n on the right hand sides of the last two equations, we arrive at the desired result (76). Similarly, we obtain (77). □
Remark 2.
For α = 1 in Theorem 18, the result reduces to
(i) 
k = 0 n n k q b k a n k H n k , q ( c ) ( b x , b y ; u ; λ ) H k , q ( c ) ( a x , a y ; u ; λ )
= k = 0 n n k q a k b n k H n k , q ( c ) ( a x , a y ; u ; λ ) H k , q ( c ) ( b x , b y ; u ; λ ) ,
(ii) 
k = 0 n n k q b k a n k H n k , q ( s ) ( b x , b y ; u ; λ ) H k , q ( s ) ( a x , a y ; u ; λ )
= k = 0 n n k q a k b n k H n k , q ( s ) ( a x , a y ; u ; λ ) H k , q ( s ) ( b x , b y ; u ; λ ) .
Remark 3.
Assume q 1 in Theorem 18, the result reduces to
(i) 
k = 0 n n k b k a n k H n k ( α , c ) ( b x , b y ; u ; λ ) H k ( α , c ) ( a x , a y ; u ; λ )
= k = 0 n n k a k b n k H n k ( α , c ) ( a x , a y ; u ; λ ) H k ( α , c ) ( b x , b y ; u ; λ ) ,
(ii) 
k = 0 n n k b k a n k H n k ( α , s ) ( b x , b y ; u ; λ ) H k ( α , s ) ( a x , a y ; u ; λ )
= k = 0 n n k a k b n k H n k ( α , s ) ( a x , a y ; u ; λ ) H k ( α , s ) ( b x , b y ; u ; λ ) .
Theorem 19.
Let a , b , > 0 with a b and n 0 . Then,
(i) 
k = 0 n n k q i = 0 a 1 j = 0 b 1 ( λ ) i + j a n k b k H n k , q ( α , c ) ( b x + b a i + j , b y ; u ; λ ) H k , q ( α , c ) ( a x , a y ; u ; λ )
= k = 0 n n k q i = 0 b 1 j = 0 a 1 ( λ ) i + j b n k a k H n k , q ( α , c ) ( a x + a b i + j , a y ; u ; λ ) H k , q ( α , c ) ( b x , b y ; u ; λ ) .
(ii) 
k = 0 n n k q i = 0 a 1 j = 0 b 1 ( λ ) i + j a n k b k H n k , q ( α , s ) ( b x + b a i + j , b y ; u ; λ ) H k , q ( α , s ) ( a x , a y ; u ; λ )
= k = 0 n n k q i = 0 b 1 j = 0 a 1 ( λ ) i + j b n k a k H n k , q ( α , s ) ( a x + a b i + j , a y ; u ; λ ) H k , q ( α , s ) ( b x , b y ; u ; λ ) .
Proof. 
Consider the identity
B ( t ) = ( 1 u ) 2 ( λ e q ( a t ) u ) ( λ e q ( b t ) u ) 2 α 1 + λ ( 1 ) a + 1 e a b z ( λ e a t + 1 ) ( λ e b z + 1 ) ( e q ( a b x t ) C O S q ( a b y t ) ) 2 .
B ( t ) = 1 u λ e q ( a z ) u α e q ( a b x t ) C O S q ( a b y t ) 1 λ ( e q ( b t ) ) a λ e q ( b t ) + 1 1 u λ e q ( b z ) u α
× 1 λ ( e q ( a z ) ) b λ e q ( a z ) + 1 e q ( a b x t ) C O S q ( a b y t )
= 1 u λ e q ( a z ) u α e q ( a b x t ) C O S q ( a b y t ) i = 0 a 1 ( λ ) i e q ( b z i ) 1 u λ e q ( b t ) u α e q ( a b x t ) C O S q ( a b y t ) j = 0 b 1 ( λ ) j e q ( a z j )
= 1 u λ e q ( a z ) u α C o s q ( a b y t ) i = 0 a 1 j = 0 b 1 ( λ ) i + j e q ( ( b x + b a i + j ) a t ) k = 0 H k , q ( α , c ) ( a x , a y ; u ; λ ) ( b t ) k [ k ] q !
= n = 0 i = 0 a 1 j = 0 b 1 ( λ ) i + j H n , q ( α , c ) ( b x + b a i + j , b y ; u ; λ ) ( a t ) n [ n ] q ! k = 0 H k , q ( α , c ) ( a x , a y ; u ; λ ) ( b t ) k [ k ] q !
= n = 0 k = 0 n n k q i = 0 a 1 j = 0 b 1 ( λ ) i + j a s k b k H n k , q ( α , c ) ( b y x + b a i + j , b η ; u ; λ )
× H k , q ( α , c ) ( a x , a y ; u ; λ ) t n [ n ] q ! .
On the other hand, we obtain
B ( t ) = n = 0 k = 0 n n k q i = 0 b 1 j = 0 a 1 ( λ ) i + j b n k a k H n k ( α , c ) ( a x + a b i + j , a y ; u ; λ )
× H k , q ( α , c ) ( b x , b y ; u ; λ ) t n [ n ] q ! .
By using (86) and (87), we arrive at the desired result (82). Similarly, we obtain (83). □
Theorem 20.
Let a , b , > 0 with a b and j 0 . Then,
k = 0 n n k q b k a n k H n k , q ( α , c ) ( b x , b y ; u ; λ ) H k , q ( α , s ) ( a x , a y ; u ; λ )
= k = 0 n n k q a k b n k H n k , q ( α , s ) ( a x , a y ; u ; λ ) H k , q ( α , c ) ( b x , b y ; u ; λ ) .
Proof. 
Suppose that
A ( t ) = ( 1 u ) 2 ( e q ( a b x t ) ) 2 C O S q ( a b y t ) S I N q ( a b y t ) ( λ e a z u ) ( λ e b z u ) 2 α .
Then, the expression for A ( t ) is symmetric in a and b, and we obtain
C ( t ) = n = 0 H n , q ( α , c ) ( b x , b y ; u ; λ ) ( a t ) n [ n ] q ! k = 0 H k , q ( α , s ) ( a x , a y ; u ; λ ) ( b t ) k [ k ] q !
= n = 0 ( k = 0 n j k q b k a j k H n k , q ( α , c ) ( b x , b y ; u ; λ ) H k , q ( α , s ) ( a x , a y ; u ; λ ) ) t n [ n ] q ! .
Similarly, we can show that
A ( z ) = n = 0 H n , q ( α , s ) ( a x , a y ; u ; λ ) ( b t ) n n ! k = 0 H k , q ( α , c ) ( b x , b y ; u ; λ ) ( a t ) k k !
= n = 0 ( k = 0 n n k q a k b n k H n k , q ( α , s ) ( a x , a y ; u ; λ ) H k , q ( α , c ) ( b x , b y ; u ; λ ) ) t n n ! .
On comparing the coefficients of t n on the right hand sides of the last two equations, we arrive at the desired result (88). □
Remark 4.
Assume that q 1 in Theorem 18, for which the result reduces to
k = 0 n n k b k a n k H n k ( α , c ) ( b x , b y ; u ; λ ) H k ( α , s ) ( a x , a y ; u ; λ )
= k = 0 n n k a k b n k H n k ( α , s ) ( a x , a y ; u ; λ ) H k ( α , c ) ( b x , b y ; u ; λ ) .

5. Symmetric Structure of Approximate Roots for q-Cosine Apostol-Type Frobenius–Euler Polynomials and Their Application

In this section, certain zeros of the q-Cosine Apostol-type Frobenius–Euler polynomials H n , q ( α , c ) ( x , y ; u ; λ ) and graphical representations are shown.
A few of them are as follows:
H 0 , q ( α , c ) ( x , y ; u ; λ ) = 1 + u u λ α , H 1 , q ( α , c ) ( x , y ; u ; λ ) = u x 1 + u u λ α u + λ + x 1 + u u λ α λ u + λ α 1 + u u λ α λ u + λ , H 2 , q ( α , c ) ( x , y ; u ; λ ) = x 2 1 + u u λ α ( 1 q ) ( 1 + q ) q 2 x 2 1 + u u λ α ( 1 q ) ( 1 + q ) y 2 1 + u u λ α ( 1 q ) ( 1 + q ) + q 2 y 2 1 + u u λ α ( 1 q ) ( 1 + q ) + u α 1 + u u λ α λ ( 1 q ) ( 1 + q ) ( u + λ ) 2 q 2 u α 1 + u u λ α λ ( 1 q ) ( 1 + q ) ( u + λ ) 2 α 1 + u u λ α λ 2 2 ( 1 q ) ( 1 + q ) ( u + λ ) 2 + q α 1 + u u λ α λ 2 2 ( 1 q ) ( 1 + q ) ( u + λ ) 2 + q 2 α 1 + u u λ α λ 2 2 ( 1 q ) ( 1 + q ) ( u + λ ) 2 q 3 α 1 + u u λ α λ 2 2 ( 1 q ) ( 1 + q ) ( u + λ ) 2 + α 2 1 + u u λ α λ 2 2 ( 1 q ) ( 1 + q ) ( u + λ ) 2 + q α 2 1 + u u λ α λ 2 2 ( 1 q ) ( 1 + q ) ( u + λ ) 2 q 2 α 2 1 + u u λ α λ 2 2 ( 1 q ) ( 1 + q ) ( u + λ ) 2 q 3 α 2 1 + u u λ α λ 2 2 ( 1 q ) ( 1 + q ) ( u + λ ) 2 x α 1 + u u λ α λ ( 1 q ) ( u + λ ) + q 2 x α 1 + u u λ α λ ( 1 q ) ( u + λ ) .
We investigate the zeros of the q-Cosine Apostol-type Frobenius–Euler polynomials H n , q ( α , c ) ( x , y ; u ; λ ) by using a computer. We plot the zeros of the q-Cosine Apostol-type Frobenius–Euler polynomials H n , q ( α , c ) ( x , y ; u ; λ ) = 0 for n = 20 (Figure 1).
In Figure 1 (top-left), we choose α = 4 , λ = 2 , u = 3 , and q = 1 10 , y = 3 . In Figure 1 (top-right), we choose α = 4 , λ = 2 , u = 3 , and q = 3 10 , y = 3 . In Figure 1 (bottom-left), we choose α = 4 , λ = 2 , u = 3 , and q = 5 10 , y = 3 . In Figure 1 (bottom-right), we choose α = 4 , λ = 2 , u = 3 , and q = 7 10 , y = 3 .
Stacks of zeros of the q-Cosine Apostol-type Frobenius–Euler polynomials H n , q ( α , c ) ( x , y ; u ; λ ) = 0 for 1 n 20 , forming a 3D structure, are presented (Figure 2).
In Figure 2 (top-left), we choose α = 4 , λ = 2 , u = 3 , and q = 1 10 , y = 3 . In Figure 2 (top-right), we choose α = 4 , λ = 2 , u = 3 , and q = 3 10 , y = 3 . In Figure 2 (bottom-left), we choose α = 4 , λ = 2 , u = 3 , and q = 5 10 , y = 3 . In Figure 2 (bottom-right), we choose α = 4 , λ = 2 , u = 3 , and q = 7 10 , y = 3 .
Next, we calculated an approximate solution satisfying the q-Cosine Apostol-type Frobenius–Euler polynomials H n , q ( α , c ) ( x , y ; u ; λ ) = 0 for q = 1 10 . The results are provided in Table 1.

6. Symmetric Structure of Approximate Roots for q-Sine Apostol-Type Frobenius–Euler Polynomials and Their Application

In this section, certain zeros of the q-Sine Apostol-type Frobenius–Euler polynomials H n , q ( α , s ) ( x , y ; u ; λ ) and beautiful graphical representations are shown.
A few of them are as follows:
H 0 , q ( α , s ) ( x , y ; u ; λ ) = 0 , H 1 , q ( α , s ) ( x , y ; u ; λ ) = y 1 + u u λ α , H 2 , q ( α , s ) ( x , y ; u ; λ ) = u x y 1 + u u λ α ( 1 q ) ( u + λ ) + q 2 u x y 1 + u u λ α ( 1 q ) ( u + λ ) + x y 1 + u u λ α λ ( 1 q ) ( u + λ ) q 2 x y 1 + u u λ α λ ( 1 q ) ( u + λ ) y α 1 + u u λ α λ ( 1 q ) ( u + λ ) + q 2 y α 1 + u u λ α λ ( 1 q ) ( u + λ ) , H 3 , q ( α , s ) ( x , y ; u ; λ ) = x 2 y 1 + u u λ α ( 1 q ) 2 ( 1 + q ) q 2 x 2 y 1 + u u λ α ( 1 q ) 2 ( 1 + q ) q 3 x 2 y 1 + u u λ α ( 1 q ) 2 ( 1 + q ) + q 5 x 2 y 1 + u u λ α ( 1 q ) 2 ( 1 + q ) q 3 y 3 1 + u u λ α ( 1 q ) 2 ( 1 + q ) ( 1 + q + q 2 ) + q 5 y 3 1 + u u λ α ( 1 q ) 2 ( 1 + q ) ( 1 + q + q 2 ) + q 6 y 3 1 + u u λ α ( 1 q ) 2 ( 1 + q ) ( 1 + q + q 2 ) q 8 y 3 1 + u u λ α ( 1 q ) 2 ( 1 + q ) ( 1 + q + q 2 ) + u y α 1 + u u λ α λ ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 q 2 u y α 1 + u u λ α λ ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 q 3 u y α 1 + u u λ α λ ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 + q 5 u y α 1 + u u λ α λ ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 y α 1 + u u λ α λ 2 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 + q y α 1 + u u λ α λ 2 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 + q 2 y α 1 + u u λ α λ 2 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2
q 4 y α 1 + u u λ α λ 2 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 q 5 y α 1 + u u λ α λ 2 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 + q 6 y α 1 + u u λ α λ 2 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 + y α 2 1 + u u λ α λ 2 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 + q y α 2 1 + u u λ α λ 2 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 q 2 y α 2 1 + u u λ α λ 2 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 q 3 y α 2 1 + u u λ α λ 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 q 4 y α 2 1 + u u λ α λ 2 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 + q 5 y α 2 1 + u u λ α λ 2 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 + q 6 y α 2 1 + u u λ α λ 2 2 ( 1 q ) 2 ( 1 + q ) ( u + λ ) 2 x y α 1 + u u λ α λ ( 1 q ) 2 ( u + λ ) + q 2 x y α 1 + u u λ α λ ( 1 q ) 2 ( u + λ ) + q 3 x y α 1 + u u λ α λ ( 1 q ) 2 ( u + λ ) q 5 x y α 1 + u u λ α λ ( 1 q ) 2 ( u + λ )
In Figure 3 (top-left), we choose α = 4 , λ = 2 , u = 3 , and q = 2 10 , y = 3 . In Figure 3 (top-right), we choose α = 4 , λ = 2 , u = 3 , and q = 4 10 , y = 3 . In Figure 3 (bottom-left), we choose α = 4 , λ = 2 , u = 3 , and q = 6 10 , y = 3 . In Figure 3 (bottom-right), we choose α = 4 , λ = 2 , u = 3 , and q = 8 10 , y = 3 .
Stacks of zeros of the q-Sine Apostol-type Frobenius–Euler polynomials H n , q ( α , s ) ( x , y ; u ; λ ) = 0 for 2 n 20 , forming a 3D structure, are presented (Figure 4).
In Figure 4 (top-left), we plot stacks of zeros of H n , q ( α , s ) ( x , y ; u ; λ ) = 0 for 2 n 30 , q = 8 10 , α = 4 , λ = 2 , u = 3 , y = 3 . In Figure 4 (top-right), we draw x and y axes but no z axis in three dimensions. In Figure 4 (bottom-left), we draw y and z axes but no x axis in three dimensions. In Figure 4 (bottom-right), we draw x and z axes but no y axis in three dimensions.
Next, we calculated an approximate solution satisfying the q-Sine Apostol-type Frobenius–Euler polynomials H n , q ( α , s ) ( x , y ; u ; λ ) = 0 for q = 8 10 . The results are given in Table 2.

7. Conclusions

By making use of q-numbers and q-concepts, Jang et al. [2,4] defined q-Bernoulli polynomials and numbers, q-Genocchi polynomials and numbers and q-Euler polynomials and numbers and provided some new and interesting identities and formulae. With this viewpoint, several authors have introduced q-analogues of special numbers and polynomials and have investigated their properties. In this paper, by making use of the q-cosine polynomials and q-sine polynomials, we have considered a new class of q-analogues of Apostol-type Frobenius–Euler polynomials and have obtained new properties and identities. In addition, we have analysed the behaviour of q-integral and q-derivative representations. Additionally, we have checked the roots and graphical representations of these polynomials by making use of Mathematica software. This approach led us to consider different methods, and special cases of used variables of newly defined polynomial in the paper. In this viewpoint, we will try to continue working on newly considered polynomials in this line.

Author Contributions

All authors contributed equally to the manuscript and written, read, and approved the final manuscript. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (No. 62172116) and the Basic Research Programs of Guizhou Province (No. QianKeHe ZK[2023]279).

Data Availability Statement

Not applicable.

Acknowledgments

The authors would like to thank the reviewers who have improved the presentation of the paper substantially.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Zeros of H n , q ( α , c ) ( x , y ; u ; λ ) .
Figure 1. Zeros of H n , q ( α , c ) ( x , y ; u ; λ ) .
Mathematics 11 02386 g001
Figure 2. Zeros of H n , q ( α , c ) ( x , y ; u ; λ ) .
Figure 2. Zeros of H n , q ( α , c ) ( x , y ; u ; λ ) .
Mathematics 11 02386 g002
Figure 3. Zeros of H n , q ( α , s ) ( x , y ; u ; λ ) .
Figure 3. Zeros of H n , q ( α , s ) ( x , y ; u ; λ ) .
Mathematics 11 02386 g003
Figure 4. Zeros of H n , q ( α , s ) ( x , y ; u ; λ ) .
Figure 4. Zeros of H n , q ( α , s ) ( x , y ; u ; λ ) .
Mathematics 11 02386 g004
Table 1. Approximate solutions of H n , q ( 4 , c ) ( x , 3 ; 3 ; 2 ) = 0 .
Table 1. Approximate solutions of H n , q ( 4 , c ) ( x , 3 ; 3 ; 2 ) = 0 .
Degree nx
1−8.0000
2−4.4000 − 4.8621 i,    −4.4000 + 4.8621 i
3−6.5429,    −1.1686 − 5.5743 i,    −1.1686 + 5.5743 i
4−5.1586 − 2.9339 i,    −5.1586 + 2.9339 i,
 0.7146 − 5.2176 i,    0.7146 + 5.2176 i
5−5.8344,    −3.3214 − 4.2756 i,    −3.3214 + 4.2756 i,
 1.7942 − 4.6626 i,    1.7942 + 4.6626 i
6−5.1069 − 2.0266 i,    −5.1069 + 2.0266 i,    −1.7739 − 4.7670 i,
 −1.7739 + 4.7670 i,    2.4363 − 4.1334 i,    2.4363 + 4.1334 i
7−5.4063,    −3.9859 − 3.2720 i,    −3.9859 + 3.2720 i,    −0.5893 − 4.8378 i,
 −0.5893 + 4.8378 i,    2.8338 − 3.6774 i,    2.8338 + 3.6774 i
8−4.9566 − 1.5254 i,    −4.9566 + 1.5254 i,    −2.8747 − 3.9790 i,
 −2.8747 + 3.9790 i,    0.2982 − 4.7116 i,    0.2982 + 4.7116 i,
 3.0887 − 3.2950 i,    3.0887 + 3.2950 i
9−5.1150,    −4.2106 − 2.6028 i,    −4.2106 + 2.6028 i,
 −1.8968 − 4.3377 i,    −1.8968 + 4.3377 i,    0.9638 − 4.5004 i,
 0.9638 + 4.5004 i,    3.2566 − 2.9754 i,    3.2566 + 2.9754 i
Table 2. Approximate solutions of H n , q ( 4 , s ) ( x , 3 ; 3 ; 2 ) = 0 .
Table 2. Approximate solutions of H n , q ( 4 , s ) ( x , 3 ; 3 ; 2 ) = 0 .
Degree nx
2−8.0000
3−7.2000 − 5.1256 i,    −7.2000 + 5.1256 i
4−9.0707,    −5.2247 − 8.2772 i,    −5.2247 + 8.2772 i
5−8.7853 − 3.8486 i,    −8.7853 + 3.8486 i,
 −3.0227 − 10.2577 i,    −3.0227 + 10.2577 i
6−9.8793,    −7.6190 − 6.7293 i,    −7.6190 + 6.7293 i,
 −0.8878 − 11.4618 i,    −0.8878 + 11.4618 i
7−9.7452 − 3.1858 i,    −9.7452 + 3.1858 i,    −6.0868 − 8.8578 i,
 −6.0868 + 8.8578 i,    1.0748 − 12.1326 i,    1.0748 + 12.1326 i
8−10.501,    −8.9527 − 5.7661 i,    −8.9527 + 5.7661i,    −4.4346 − 10.3981 i,
 −4.4346 + 10.3981 i,    2.8319 − 12.4327 i,    2.8319 + 12.4327 i
9−10.4252 − 2.7531 i,    −10.4252 + 2.7531 i,    −7.8126 − 7.8189 i,
 −7.8126 + 7.8189 i,    −2.7879 − 11.4801 i,    −2.7879 + 11.4801 i,
 4.3811 − 12.4748 i,    4.3811 + 12.4748 i
10−10.985,    −9.8411 − 5.0793 i,    −9.8411 + 5.0793 i,
 −6.5042 − 9.4234 i,    −6.5042 + 9.4234 i,    −1.2119 − 12.2069 i,
 −1.2119 + 12.2069 i,    5.7340 − 12.3389 i,    5.7340 + 12.3389 i
11−10.9347 − 2.4373 i,    −10.9347 + 2.4373 i,    −8.9518 − 7.0112 i,
 −8.9518 + 7.0112 i,    −5.1343 − 10.6536 i,    −5.1343 + 10.6536 i,
 0.2605 − 12.6598 i,    0.2605 + 12.6598 i,    6.9078 − 12.0821 i,
 6.9078 + 12.0821 i
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Rao, Y.; Khan, W.A.; Araci, S.; Ryoo, C.S. Explicit Properties of Apostol-Type Frobenius–Euler Polynomials Involving q-Trigonometric Functions with Applications in Computer Modeling. Mathematics 2023, 11, 2386. https://doi.org/10.3390/math11102386

AMA Style

Rao Y, Khan WA, Araci S, Ryoo CS. Explicit Properties of Apostol-Type Frobenius–Euler Polynomials Involving q-Trigonometric Functions with Applications in Computer Modeling. Mathematics. 2023; 11(10):2386. https://doi.org/10.3390/math11102386

Chicago/Turabian Style

Rao, Yongsheng, Waseem Ahmad Khan, Serkan Araci, and Cheon Seoung Ryoo. 2023. "Explicit Properties of Apostol-Type Frobenius–Euler Polynomials Involving q-Trigonometric Functions with Applications in Computer Modeling" Mathematics 11, no. 10: 2386. https://doi.org/10.3390/math11102386

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