Home Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
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Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions

  • Cristina B. Corcino , Wilson D. Castañeda and Roberto B. Corcino EMAIL logo
Published/Copyright: August 24, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

The study of special functions has become an enthralling area in mathematics because of its properties and wide range of applications that are relevant into other fields of knowledge. Developing topics in special functions involves the investigation of Apostol-type polynomials encompassing the combinations, extensions, and generalizations of some classical polynomials such as Bernoulli, Euler, Genocchi, and tangent polynomials. One particular type of these polynomials is the Apostol-Frobenius-Euler polynomials of order a denoted by H n α ( z ; u ; λ ) . Using the saddle point method, Corcino et al. obtained approximations for the higher-order tangent polynomials. They also established a new method to derive its approximations with enlarged region of validity. In this article, it is found that these methods are applicable to the higher-order Apostol-Frobenius-Euler polynomials. Consequently, approximations of higher-order Apostol-Frobenius-Euler polynomials in terms of the hyperbolic functions are obtained for large values of the parameter n , and its uniform approximations with enlarged region of validity are also derived. Moreover, approximations of the generalized Apostol-type Frobenius-Euler polynomials of order α with parameters a , b , and c are obtained by applying the same methods. Graphs are provided to show the accuracy of the exact values of these polynomials and their corresponding approximations for some specific values of the parameters.

Keywords: Apostol-Frobenius-Euler polynomials; generalized Apostol-type Froebnius-Euler polynomials; Euler Polynomials; Hermite polynomials; asymptotic approximation
MSC 2010: 11B68; 11B83; 41A60

1 Introduction

Recent developments in the study of special functions have been constructed and investigated by mixing the concepts of well-known special polynomials such as Bernoulli, Euler, Genocchi, tangent polynomials, Gegenbauer polynomials, Hermite polynomials, and Laguerre polynomials [1–3]. An interesting combination of special polynomials that can be established and explored is through merging the concept of Euler polynomials with the Apostol and Frobenius polynomials.

The Apostol-Frobenius-Euler polynomials of order α denoted by H n α ( z ; u ; λ ) are defined by the generating function [4,5]

(1) 1 − u λ e w − u α e z w = ∑ n = 0 ∞ H n α ( z ; u ; λ ) w n n ! , ∣ w ∣ < log λ u ,

where λ , u ∈ C with λ ≠ 0 , u ≠ 1 , and α ∈ Z .

When λ = 1 , this gives the generating function of the Frobenius-Euler polynomials of order α given as [6,7]

(2) 1 − u e w − u α e z w = ∑ n = 0 ∞ H n α ( z ; u ) w n n ! , ∣ w ∣ < ∣ log ( u ) ∣ .

When u = − 1 , this gives the generating function of the Apostol-Euler polynomials of order α defined as [8]

(3) 2 λ e w + 1 α e z w = ∑ n = 0 ∞ ℰ n α ( z ; λ ) w n n ! , ∣ w ∣ < ∣ π i − log ( λ ) ∣ .

Observe that setting z = 0 in (1) and (2),

(4) H n α ( 0 ; u ; λ ) = H n α ( u ; λ ) and H n α ( 0 ; u ) = H n α ( u ) ,

where H n α ( u ; λ ) and H n α ( u ) are called the Apostol-Frobenius-Euler numbers of order α and the Frobenius-Euler numbers of order α , respectively [9,10].

In the case when α = 1 in (1) and (2),

(5) H n 1 ( z ; u ; λ ) = H n ( z ; u ; λ ) and H n 1 ( z ; u ) = H n ( z ; u ) ,

where H n ( z ; u ; λ ) and H n ( z ; u ) are called the Apostol-Frobenius-Euler polynomials and the Frobenius-Euler polynomials, respectively (see [11,12]).

The first few values of the Apostol-Frobenius-Euler polynomials of order α are determined explicitly as follows:

H 0 α ( z ; u ; λ ) = 1 − u λ − u α , H 1 α ( z ; u ; λ ) = − 1 − u λ − u α u z + ( α − z ) λ ( λ − u ) , H 2 α ( z ; u ; λ ) = 1 − u λ − u α u 2 z 2 + u ( α + 2 α z − 2 z 2 ) λ + ( α − z ) 2 λ 2 ( λ − u ) 2 , H 3 α ( z ; u ; λ ) = − 1 − u λ − u α λ [ u 3 z 3 + u 2 ( α − 3 z 3 + 3 α z ( 1 + z ) ) ] ( λ − u ) 3 − 1 − u λ − u α u [ α + 3 z 3 + 3 α 2 ( 1 + z ) − 3 α z ( 1 + 2 z ) ] ( λ − u ) 3 , and H 4 α ( z ; u ; λ ) = 1 − u λ − u α λ [ u 4 z 4 + u 3 ( α − 4 z 4 + 2 α z ( 2 + z ( 3 + 2 z ) ) ) ] ( λ − u ) 4 1 − u λ − u α u 2 [ 6 z 4 − 4 α ( − 1 + 3 z 2 ( 1 + z ) ) ] ( λ − u ) 4 .

The Apostol-Frobenius-Euler polynomials of order α H n α ( z ; u ; λ ) are λ extensions of the Frobenius-Euler polynomials. Frobenius-Euler polynomials and numbers are named after the great German mathematician Ferdinand Georg Frobenius [13], who made essential works on the context of these polynomials in number theory and the relation of their divisibility properties with the Stirling numbers of the second kind [14,15].

Analogues and other extensions of the Apostol-type Frobenius-Euler polynomials play a significant role in the development of some concepts and applications in calculus, differential equations, number theory, and physics [16–22]. In addition, the Fourier expansions and integral representations of these polynomials are given in [11,23,24].

In the study of Corcino et al. [25–27], uniform approximations for the higher-order tangent polynomials were derived using the saddle point method. In addition, they also presented a new method to obtain its asymptotic expansions with enlarge region of validity. However, the asymptotic approximations of higher-order Apostol-Frobenius-Euler polynomials which are parallel to the results in [27] are not mentioned and provided in other related studies.

In this study, the asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α are derived using the methods employed in [27–29]. Moreover, asymptotic expansion of the generalized Apostol-type Frobenius-Euler polynomials of order α with parameters a , b , and c , which is denoted by ℋ n α ( z ; u ; a , b , c , λ ) is also obtained. Asymptotic formulas for the special cases and different values of the parameters are also given as corollaries.

2 Uniform approximations

In this section, the main results of the study explore the uniform approximation of the Apostol-Frobenius-Euler polynomials of order α . Using the saddle point method, the following theorem contains the derived approximation.

Theorem 2.1

For n , α ∈ Z + , u , λ ∈ C \ { 0 , 1 } , and z ∈ C \ { 0 } such that ∣ Im z − 1 ∣ < 2 π − Arg λ u or ∣ z − 1 ∣ < ∣ z − 1 − ( 2 π i − δ ) ∣ , the Apostol-Frobenius-Euler polynomials of order α satisfy

(6) H n α n z + α 2 ; u ; λ = ( n z ) n 1 − u 2 λ u α csch α δ z + 1 2 z 1 − α α + ( α + 1 ) csch 2 δ z + 1 2 z 8 n z 2 + O 1 n 2 ,

where δ = log λ u and the logarithm is taken to be the principal branch.

Proof

Applying the Cauchy integral formula to (1)

(7) H n α ( z ; u ; λ ) = n ! 2 π i ∫ C ( 1 − u ) α e z w ( λ e w − u ) α d w w n + 1 ,

where C is a circle about 0 with radius lesser than ∣ 2 π i − ln λ u ∣ . Writing δ = log λ u , (7) can be expressed as

H n α ( z ; u ; λ ) = n ! 2 π i ∫ C 1 − u u α e z w ( e δ + w − 1 ) α d w w n + 1 .

With 2 e ( δ + w ) 2 sinh δ + w 2 α = ( e δ + w − 1 ) α , it follows that

H n α ( z ; u ; λ ) = n ! 2 π i 1 − u 2 λ u α ∫ C f ( w ) e z w e α w 2 d w w n + 1 ,

where f ( w ) = 1 ⁄ sinh α δ + w 2 = csch α δ + w 2 . It can be observed that the function f ( w ) is a meromorphic function with simple poles of order α at the zeros of sinh δ + w 2 , which are given by w j = 2 j π i − δ , j = ± 1 , ± 2 , … .

It follows that by taking z → n z and letting n z → ∞ with fixed z ,

(8) H n α n z + α 2 ; u ; λ = n ! 2 π i 1 − u 2 λ u α ∫ C f ( w ) e n ( z w − log w ) d w w .

The main contribution of the integrand to the integral in (8) originates at the saddle point of the argument of the exponential [30]. Thus, if z − 1 is not a pole, then the approximations of H n α n z + α 2 ; u ; λ can be derived by expanding f ( w ) around the saddle point w = z − 1 . It follows from Lemmas 1 and 2 and Theorem 1 of [31] that

(9) H n α n z + α 2 ; u ; λ = ( n z ) n 1 − u 2 λ u α ∑ k = 0 ∞ f ( k ) ( z − 1 ) k ! p k ( n ) ( n z ) k ,

where p k ( n ) are the polynomials

(10) p 0 ( n ) = 1 , p 1 ( n ) = 0 , p 2 ( n ) = − n , p 3 ( n ) = 2 n ,

(11) p k ( n ) = ( 1 − k ) p k − 1 ( n ) + n p k − 2 ( n ) , k ≥ 3 .

Expanding the sum in (9) and keeping only the first three terms yield (6) (Figure 1).

Figure 1 Solid lines represent the Apostol-Frobenius-Euler polynomials of order α H n α ( n z + α 2 ; u ; λ ) \alpha {H}_{n}^{\alpha }\left(nz+\frac{\alpha }{2};\hspace{0.33em}u;\hspace{0.33em}\lambda ) for several values of n n , whereas dashed lines represent the right-hand side of (6) with z ≡ x z\equiv x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 {\left(1+{| \frac{x}{\sigma }| }^{n})}^{-1} where we choose σ = 0.5 \sigma =0.5 . (a) n = 8 , α = 7 , u = − 3 , λ = 4 , b = 0.5 n=8,\alpha =7,u=-3,\lambda =4,b=0.5 and (b) n = 5 , α = 4 , u = 3 , λ = 7 , b = 0.5 n=5,\alpha =4,u=3,\lambda =7,b=0.5 .
Figure 1

Solid lines represent the Apostol-Frobenius-Euler polynomials of order α H n α ( n z + α 2 ; u ; λ ) for several values of n , whereas dashed lines represent the right-hand side of (6) with z ≡ x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 where we choose σ = 0.5 . (a) n = 8 , α = 7 , u = − 3 , λ = 4 , b = 0.5 and (b) n = 5 , α = 4 , u = 3 , λ = 7 , b = 0.5 .

The following corollaries give the uniform approximations of the Frobenius-Euler polynomials of order α , Apostol-Euler polynomials of order α , and Euler polynomials of order α .

Corollary 2.2

For n , α ∈ Z + , u ∈ C \ { 0 , 1 } , and z ∈ C \ { 0 } such that ∣ Im z − 1 ∣ < 2 π − Arg 1 u or ∣ z − 1 ∣ < ∣ z − 1 − ( 2 π i + ν ) ∣ , the Frobenius-Euler polynomials of order α satisfy

(12) H n α n z + α 2 ; u = ( n z ) n ( 1 − u ) α ( 2 u ) α csch α 1 − z ν 2 z 1 − α α + ( α + 1 ) csch 2 1 − z ν 2 z 8 n z 2 + O 1 n 2 ,

where ν = log ( u ) and the logarithm is taken to be the principal branch.

Proof

This follows from Theorem 2.1 by taking λ = 1 .□

Corollary 2.3

For n , α ∈ Z + , λ ∈ C \ { 0 , 1 } , and z ∈ C \ { 0 } such that ∣ Im z − 1 ∣ < π − Arg ( λ ) or ∣ z − 1 ∣ < ∣ z − 1 − ( π i − τ ) ∣ , the Apostol-Euler polynomials of order α satisfy

(13) ℰ n α n z + α 2 ; λ = ( n z ) n ( λ ) α sech α 1 + z τ 2 z 1 − α α − ( α + 1 ) sech 2 1 − z τ 2 z 8 n z 2 + O 1 n 2 ,

where τ = log ( λ ) and the logarithm is taken to be the principal branch.

Proof

This follows from Theorem 2.1 by taking u = − 1 .□

Corollary 2.4

For n , α ∈ Z + and z ∈ C \ 0 such that ∣ Im z − 1 ∣ < π or ∣ z − 1 ∣ < ∣ z − 1 − π ∣

(14) E n α n z + α 2 = ( n z ) n sech α 1 2 z 1 − α α − ( α + 1 ) sech 2 1 2 z 8 n z 2 + O 1 n 2 .

Proof

This follows from 2.1 by taking λ = 1 and u = − 1 .□

Remark 2.5

Taking α = 1 in (14), approximation for the classical Euler polynomials E n ( n z + 1 2 ) is obtained similar to that of Lopez and Temme (Corollary 2, [31]).

Figures 2, 3, and 4 show the accuracy of the asymptotic formulae obtained in Corollaries (12), (13), and (14), respectively.

Figure 2 Solid lines represent the Frobenius-Euler polynomials of order α H n α ( n z + α 2 ; u ) \alpha {H}_{n}^{\alpha }\left(nz+\frac{\alpha }{2};\hspace{0.33em}u) for several values of n n , whereas dashed lines represent the right-hand side of (12) with z ≡ x z\equiv x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 {\left(1+{| \frac{x}{\sigma }| }^{n})}^{-1} where we choose σ = 0.5 \sigma =0.5 . (a) n = 4 , α = 3 , u = 5 n=4,\alpha =3,u=5 and (b) n = 9 , α = 4 , u = − 2 n=9,\alpha =4,u=-2 .
Figure 2

Solid lines represent the Frobenius-Euler polynomials of order α H n α ( n z + α 2 ; u ) for several values of n , whereas dashed lines represent the right-hand side of (12) with z ≡ x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 where we choose σ = 0.5 . (a) n = 4 , α = 3 , u = 5 and (b) n = 9 , α = 4 , u = − 2 .

Figure 3 Solid lines represent the Apostol-Euler polynomials of order α ℰ n α ( n z + α 2 , λ ) \alpha {{\mathcal{ {\mathcal E} }}}_{n}^{\alpha }\left(nz+\frac{\alpha }{2},\lambda ) for several values of n n , whereas dashed lines represent the right-hand side of (13) with z ≡ x z\equiv x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 {\left(1+{| \frac{x}{\sigma }| }^{n})}^{-1} where we choose σ = 0.5 \sigma =0.5 . (a) n = 5 , α = 6 , λ = 4 n=5,\alpha =6,\lambda =4 and (b) n = 6 , α = 4 , λ = 8 n=6,\alpha =4,\lambda =8 .
Figure 3

Solid lines represent the Apostol-Euler polynomials of order α ℰ n α ( n z + α 2 , λ ) for several values of n , whereas dashed lines represent the right-hand side of (13) with z ≡ x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 where we choose σ = 0.5 . (a) n = 5 , α = 6 , λ = 4 and (b) n = 6 , α = 4 , λ = 8 .

Figure 4 Solid lines represent the Euler polynomials of order α E n α ( n z + α 2 ) \alpha {E}_{n}^{\alpha }\left(nz+\frac{\alpha }{2}) for several values of n n , whereas dashed lines represent the right-hand side of (14) with z ≡ x z\equiv x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 {\left(1+{| \frac{x}{\sigma }| }^{n})}^{-1} where we choose σ = 0.5 \sigma =0.5 . (a) n = 5 , α = 4 , b = 0.5 n=5,\alpha =4,b=0.5 and (b) n = 7 , α = 8 , b = 0.5 n=7,\alpha =8,b=0.5 .
Figure 4

Solid lines represent the Euler polynomials of order α E n α ( n z + α 2 ) for several values of n , whereas dashed lines represent the right-hand side of (14) with z ≡ x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 where we choose σ = 0.5 . (a) n = 5 , α = 4 , b = 0.5 and (b) n = 7 , α = 8 , b = 0.5 .

3 Enlarged region of validity

The previous section contains an approximation valid in the region ∣ z − 1 ∣ < ∣ z − − w j ∣ with poles w j = ± 1 , ± 2 , … . This region may be enlarged by isolating the contribution of the poles of f ( w ) . In this section, an asymptotic expansion with an enlarged region of validity is obtained in the following theorem.

Theorem 3.1

For λ , u ∈ C \ { 0 , 1 } , α ∈ Z + , and z ∈ C such that ∣ z − 1 ∣ < ∣ z − 1 − w k ∣ for all k = l + 1 , l + 2 , … , the Apostol-Frobenius-Euler polynomials of order α satisfy

(15) H n α n z + α 2 ; u ; λ = ( 1 − u ) α ( 2 λ u ) α ∑ k = 1 l ∑ j = 1 α e w k n z r k j ∑ s = 0 n n s ( − 1 ) ( j − 1 ) ⟨ j − 1 ⟩ s ( w k ) − ( j − 1 + s ) ( n − s ) ! w k n − s + 1 − Γ ( n − s + 1 , w k n z ) w k n − s + 1 + ( − 1 ) j ⟨ j ⟩ n w k j + n + ( n z ) n ∑ k = 0 ∞ f ( k ) ( z − 1 ) − h l ( k ) ( z − 1 ) k ! p k ( n ) ( n z ) k ,

where the polynomials p k ( n ) are given in (11) and h l k is the kth derivative of the function h l ( w ) given by (28) and

∑ j = 1 m r k j ( w − w k ) j

are the given principal parts of the Laurent series corresponding to the poles w k = 2 k π − δ , where δ = log λ u and the entire function h l ( w ) is determined by f ( w ) = csch α δ + w 2 .

Proof

Using Mittag-Leffler’s theorem [32,33], write f ( w ) = cosh α δ + w 2 as

(16) f ( w ) = ∑ k = 1 l ∑ j = 1 m r k j ( w − w k ) j + q k ( w ) + g ( w ) = ∑ k = 1 l ∑ j = 1 α r k j ( w − w k ) j + f l ( w ) ,

where

(17) f l ( w ) = ∑ k = 1 l q k ( w ) + g ( w ) ,

q k ( w ) is a polynomial of w , r k j are residues at w k , k = 1 , 2 , … , l . With this, f l ( w ) has no poles inside the disk ∣ w ∣ < ∣ w m + 1 ∣ . Recall from (8),

(18) H n α n z + α 2 ; u ; λ = n ! 2 π i ( 1 − u ) α ( 2 λ u ) α ∫ C f ( w ) e w n z d w w n + 1 ,

where f ( w ) = 1 ⁄ sinh α δ + w 2 = csch α δ + w 2 . Substituting (16) into (18) gives

(19) H n α n z + α 2 ; u ; λ = n ! 2 π i ( 1 − u ) α ( 2 λ u ) α ∫ C ∑ k = 1 l ∑ j = 1 α r k j ( w − w k ) j + f l ( w ) e w n z d w w n + 1 = X l n , α ( z ) + Y l n , α ( z ) ,

where

(20) X l n , α ( z ) = n ! 2 π i ( 1 − u ) α ( 2 λ u ) α ∫ C f l ( w ) e w n z d w w n + 1 ,

(21) Y l n , α ( z ) = ( 1 − u ) α ( 2 λ u ) α ∫ C n ! 2 π i ∑ k = 1 l ∑ j = 1 α r k j ( w − w k ) j e w n z d w w n + 1 .

To evaluate (20), repeat the process used in the last section for f l ( w ) instead of f ( w ) , thus we have

(22) X l n , α ( z ) = n ! 2 π i ( 1 − u ) α ( 2 λ u ) α ∫ C f l ( w ) e n ( w z − log w ) d w w .

The approximations of X l n , α ( z ) can be obtained by expanding f l ( w ) around the saddle point z − 1 . Assume z − 1 is not a pole of f l ( w ) , we can expand it as

(23) f l ( w ) = ∑ k = 0 ∞ f l k ( z − 1 ) k ! ( w − z − 1 ) k , ∣ w − z − 1 ∣ < r ,

where r is the distance from z − 1 to the nearest singularity of f l ( w ) . Introducing (23) into (20)

(24) X l n , α ( z ) = ( n z ) n ( 1 − u ) α ( 2 λ u ) α ∑ k = 0 ∞ s f l k ( z − 1 ) k ! u k ( n , z ) ,

where

(25) u k ( n , z ) = 1 ( n z ) n n ! 2 π i ∫ C ( w − z − 1 ) k e w n z d w w n + 1 .

Note that the functions (25) can be represented as (Lemma 1, [31])

(26) u k ( n , z ) = p k ( n ) ( n z ) k ,

where p k ( n ) are the polynomials in (11). Thus, (20) may be expanded as the infinite sum

(27) X l n , α ( z ) = ( n z ) n ( 1 − u ) α ( 2 λ u ) α ∑ k = 0 ∞ f l k ( z − 1 ) k ! p k ( n ) ( n z ) k ,

valid for α ∈ Z + , z ∈ C \ { 0 } such that ∣ z − 1 ∣ < ∣ z − 1 − w j ∣ for j = l + 1 , l + 2 , … given the first 2 l poles of f ( w ) . It follows from (16) that the k th derivative of f l ( w ) is

f l ( k ) ( w ) = f ( k ) ( w ) − h l ( k ) ( w ) ,

where

(28) h l ( w ) = − ∑ k = 1 l ∑ j = 1 α r k j ( w − w k ) j .

Thus, the expansion of X l n , α ( z ) in (20) is

(29) X l n , α ( z ) = ( n z ) n ( 1 − u ) α ( 2 λ u ) α ∑ k = 0 ∞ f l ( k ) ( z − 1 ) − h l ( k ) ( z − 1 ) k ! p k ( n ) ( n z ) k

valid for ∣ z − 1 ∣ < ∣ z − 1 − w j ∣ , j = l + 1 , l + 2 , … and z ≠ 0 . The expansion’s range of validity is larger than that of the expansion in Theorem (2.1).

Conversely, to obtain an expansion for Y l n , α ( z ) , we employ similar computations from Corcino et al. [27]. Shifting the integration contour by w = w k + t in each integral in (21), it follows that d w = d t and

(30) Y l n , α ( z ) = ( 1 − u ) α ( 2 λ u ) α ∑ k = 1 l ∑ j = 1 α e w k n z r k j n ! 2 π i ∫ C ′ e t n z t j d t ( w k + t ) n + 1 ,

where C ′ : t = − w k + R e i θ , − π < θ ≤ π is a circle with radius R and center at − w k . Note that 0 is not on the w ′ s . This C ′ is the image of C : w = R e i θ through the shift w = w k + t . To continue, we use the idea that

(31) e t n z t j = ∫ 0 n z e t x t j − 1 + 1 t j .

Substituting (31) into (30), we have

(32) Y l n , α ( z ) = ( 1 − u ) α ( 2 λ u ) α ∑ k = 1 l ∑ j = 1 α e w k n z r k j n ! 2 π i ∫ C ′ ∫ 0 n z e t x t j − 1 + 1 t j d w ( w k + t ) n + 1 .

The proceeding parts are expositions of the computations from [27]. Now, we determine

(33) n ! 2 π i ∫ C ′ e t x t j − 1 d w ( w k + t ) n + 1 = d n d t n e t x t − ( j − 1 ) t = − w k .

Using the Leibniz rule for differentiation, (33) can be written as

(34) n ! 2 π i ∫ C ′ e t x t j − 1 d w ( w k + t ) n + 1 = ∑ s = 0 n n s x n − s e − w k x ( − 1 ) ( j − 1 ) ⟨ j − 1 ⟩ s ( w k ) − ( j − 1 + s ) ,

where ⟨ j − 1 ⟩ s denote the rising factorial of j − 1 with increment s . It can also be computed that

(35) n ! 2 π i ∫ C ′ t − j d w ( w k + t ) n + 1 = d n d t n ( t − j ) t = w k = ( − 1 ) j ⟨ j ⟩ n w k j + n .

Consider the incomplete gamma function

(36) Γ ( n − s + 1 , w k z ) = ∫ w k z ∞ e − t t n − s d t .

Let η = t w k . Then t = η w k and w k d η = d t . Moreover, t = ∞ ⇔ η = ∞ ; t = w k z ⇔ η = z . Thus, (36) becomes

(37) Γ ( n − s + 1 , w k z ) = ∫ z ∞ e − w k η ( w k η ) n − s w k d η .

From this, it can be shown that

(38) ∫ 0 z e − w k η η n − s d η = ∫ 0 ∞ e − w k η η n − s d η − Γ ( n − s + 1 , w k z ) w k n − s + 1 .

Note that z ↦ n z . Then

(39) ∫ 0 n z e − w k η η n − s d η = ∫ 0 ∞ e − w k η η n − s d η − Γ ( n − s + 1 , w k n z ) w k n − s + 1 .

Substituting (34) and (35) into (32) gives

(40) Y l n , α ( z ) = ( 1 − u ) α ( 2 λ u ) α ∑ k = 1 l ∑ j = 1 α e w k n z r k j ∫ 0 n z ∑ s = 0 n n s x n − s e − w k x ( − 1 ) ( j − 1 ) ⟨ j − 1 ⟩ s ( w k ) − ( j − 1 + s ) d x + ( − 1 ) j ⟨ j ⟩ n w k j + n = ( 1 − u ) α ( 2 λ u ) α ∑ k = 1 l ∑ j = 1 α e w k n z r k j ∑ s = 0 n n s ( − 1 ) ( j − 1 ) ⟨ j − 1 ⟩ s ( w k ) − ( j − 1 + s ) ∫ 0 n z x n − s e − w k x d x + ( − 1 ) j ⟨ j ⟩ n w k j + n .

Using (39) in (40), we obtain

(41) Y l n , α ( z ) = ( 1 − u ) α ( 2 λ u ) α ∑ k = 1 l ∑ j = 1 α e w k n z r k j ∑ s = 0 n n s ( − 1 ) ( j − 1 ) ⟨ j − 1 ⟩ s ( w k ) − ( j − 1 + s ) × ∫ 0 ∞ e − w k t t n − s d t − Γ ( n − s + 1 , w k n z ) w k n − s + 1 + ( − 1 ) j ⟨ j ⟩ n w k j + n .

Note that for n ≥ s ,

(42) ∫ 0 ∞ t n − s e − w k t d t = ( n − s ) ! w k n − s + 1 .

Hence, (41) can be written as

(43) Y l n , α ( z ) = ( 1 − u ) α ( 2 λ u ) α ∑ k = 1 l ∑ j = 1 α e w k n z r k j ∑ s = 0 n n s ( − 1 ) ( j − 1 ) ⟨ j − 1 ⟩ s ( w k ) − ( j − 1 + s ) ( n − s ) ! w k n − s + 1 − Γ ( n − s + 1 , w k n z ) w k n − s + 1 + ( − 1 ) j ⟨ j ⟩ n w k j + n .

Substituting (32) and (43) into (21), we have

H n α n z + α 2 ; u ; λ = ( 1 − u ) α ( 2 λ u ) α ∑ k = 1 l ∑ j = 1 α e w k n z r k j ∑ s = 0 n n s ( − 1 ) ( j − 1 ) ⟨ j − 1 ⟩ s ( w k ) − ( j − 1 + s ) ( n − s ) ! w k n − s + 1 − Γ ( n − s + 1 , w k n z ) w k n − s + 1 + ( − 1 ) j ⟨ j ⟩ n w k j + n + ( n z ) n ∑ k = 0 ∞ f ( k ) ( z − 1 ) − h l ( k ) ( z − 1 ) k ! p k ( n ) ( n z ) k

valid for α ∈ Z + such that for all k = l + 1 , l + 2 , … , where the polynomials p k ( n ) are given in (11) and h l ( k ) is the k th derivative of h l ( w ) given by (28).□

The accuracy of the asymptotic formula obtained in (6) and (15) is shown in Figure 5.

Figure 5 Solid lines in (a) and (b) represent H n α ( n z + α 2 ; u ; λ ) {H}_{n}^{\alpha }\left(nz+\frac{\alpha }{2};\hspace{0.33em}u;\hspace{0.33em}\lambda ) , whereas dashed lines in (a) and (b) represent the right-hand side of (6) and (15) for n = 3 , α = 3 , u = 4 , n=3,\alpha =3,u=4, and λ = 6 \lambda =6 , respectively, with z ≡ x z\equiv x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 {\left(1+{| \frac{x}{\sigma }| }^{n})}^{-1} where we choose σ = 0.5 \sigma =0.5 .
Figure 5

Solid lines in (a) and (b) represent H n α ( n z + α 2 ; u ; λ ) , whereas dashed lines in (a) and (b) represent the right-hand side of (6) and (15) for n = 3 , α = 3 , u = 4 , and λ = 6 , respectively, with z ≡ x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 where we choose σ = 0.5 .

The following corollary gives the approximation with enlarged region of validity for the Frobenius-Euler polynomials.

Corollary 3.2

For u ∈ C \ { 0 , 1 } , α , n ∈ Z + , and z ∈ C such that ∣ z − 1 ∣ < ∣ z − 1 − w k ∣ for all ∣ z − 1 ∣ for all k = l + 1 , l + 2 , … , the Frobenius-Euler polynomials of order α satisfy

(44) H n α n z + α 2 ; u = ( 1 − u ) α ( 2 u ) α ∑ k = 1 l ∑ j = 1 α e w k n z r k j ∑ s = 0 n n s ( − 1 ) ( j − 1 ) ⟨ j − 1 ⟩ s ( w k ) − ( j − 1 + s ) ( n − s ) ! w k n − s + 1 − Γ ( n − s + 1 , w k n z ) w k n − s + 1 + ( − 1 ) j ⟨ j ⟩ n w k j + n + ( n z ) n ∑ k = 0 ∞ f ( k ) ( z − 1 ) − h l ( k ) ( z − 1 ) k ! p k ( n ) ( n z ) k ,

where the polynomials p k ( n ) are given in (11) and h l k is the kth derivative of the function h l ( w ) given by (28) and

∑ j = 1 m r k j ( w − w k ) j ,

are the given principal parts of the Laurent series corresponding to the poles w k = 2 π i + ν , where ν = log ( u ) and the entire function h l ( w ) is determined by f ( w ) = csch α w − ν 2 .

Proof

This follows from Theorem 3.1 by taking λ = 1 .□

The following table shows the exact and approximate values of the Apostol-Frobenius-Euler polynomials of order α , H n α ( n z + α ⁄ 2 ; u ; λ ) , where n = 4 , α = 3 , u = 2 , and λ = 5 for some increasing values of z = x .

It can be observed from Table 1 that for increasing values of z , the relative error decreases.

Table 1

Exact and approximate values of H 4 3 ( 4 z + 3 ⁄ 2 ; 2 ; 5 )

z Exact values H 4 3 ( 4 z + 3 ⁄ 2 ; 2 ; 5 ) Approximate values using Theorem 2.1 Relative error
3 ‒ 2.39241 × 1 0 2 ‒ 2.42462 × 1 0 2 1.34628 × 1 0 ‒ 2
4 ‒ 1.00774 × 1 0 3 ‒ 1.01673 × 1 0 3 8.91731 × 1 0 ‒ 3
5 ‒ 2.93002 × 1 0 3 ‒ 2.94714 × 1 0 3 5.84244 × 1 0 ‒ 3
6 ‒ 6.83097 × 1 0 3 ‒ 6.8580 × 1 0 3 3.9568 × 1 0 ‒ 3
7 ‒ 1.3763 × 1 0 4 ‒ 1.38013 × 1 0 4 2.78004 × 1 0 ‒ 3
8 ‒ 2.50062 × 1 0 4 ‒ 2.50567 × 1 0 4 2.01911 × 1 0 ‒ 3
9 ‒ 4.2068 × 1 0 4 ‒ 4.21315 × 1 0 4 1.50901 × 1 0 ‒ 3
10 ‒ 6.66836 × 1 0 4 ‒ 6.67607 × 1 0 4 1.15571 × 1 0 ‒ 3
11 ‒ 1.00816 × 1 0 5 ‒ 1.00907 × 1 0 5 9.0386 × 1 0 ‒ 4
12 ‒ 1.46654 × 1 0 5 ‒ 1.4676 × 1 0 5 7.19781 × 1 0 ‒ 4

4 Generalized Apostol-type Frobenius-Euler polynomials

For parameters λ , u , ∈ C , u ≠ λ and a , b , c ∈ R + with a ≠ b , the generalized Apostol-type Frobenius-Euler polynomials of order α , denoted by ℋ n α ( z ; u ; a , b , c , λ ) , are defined by means of the following generating function [24]:

(45) a w − u λ b w − u α c z w = ∑ n = 0 ∞ ℋ n α ( z ; u ; a , b , c , λ ) w n n ! , ∣ w ∣ < log λ u ln b .

On setting a = 1 , b = e m , c = e , and λ = 1 in (45), Belbachir and Souddi [34] gives the generalized Frobenius-Euler polynomials of order α with parameter m , ℋ n α ( z ; u ; m ) , defined by the following generating function:

(46) 1 − u e m w − u α c z w = ∑ n = 0 ∞ ℋ n α ( z ; u ; m ) w n n ! , ∣ w ∣ < 2 π .

When a = 1 , b = e m , c = e , u = − 1 , and λ = 1 in (45), this results in the generalized Euler polynomials of order α with parameter m , E n a ( z ; m ) , defined by the generating function [34]

(47) 2 e m w + 1 α e z w = ∑ n = 0 ∞ E n α ( z ; m ) w n n ! , ∣ w ∣ < 2 π .

In this section, the methods used to obtain the approximations in Theorem 2.1 and 3.1 are applied to the case of the generalized Apostol-type Frobenius-Euler polynomials of order α with parameters a , b , and c . Approximations for the generalized Frobenius-Euler polynomials and generalized Euler polynomials of order α with parameter m are given as corollaries.

4.1 Uniform expansions

Uniform approximations for the generalized Apostol-type Frobenius-Euler polynomials are obtained using the saddle point method. The following theorem describes the said approximation.

Theorem 4.1

For n , α ∈ Z + , a , b , c ∈ R + , u , λ ∈ C \ { 0 , 1 } , and z ∈ C \ { 0 } such that ∣ Im z − 1 ∣ < 2 π β − Arg δ β or ∣ z − 1 ∣ < ∣ z − 1 − 2 π i − δ β ∣ , the generalized Apostol-type Frobenius-Euler polynomials of order α satisfy

(48) ℋ n α n z γ + α 2 ; u ; a , b , c , λ = ( n z ) n c b α 2 z ( a 1 ⁄ z − u ) α ( 2 λ u ) α sinh α z δ + β 2 z 1 − α 2 n z 2 log ( a ) a 1 ⁄ z ( a 1 ⁄ z − u ) log ( a ) + α log c b − α β coth z δ + β 2 z + α β 2 coth z δ + β 2 z β 2 coth z δ + β 2 z − log c b + 1 4 β 2 csch 2 z δ + β 2 z + α log 2 c b + O 1 n 2 ,

where δ = log λ u , β = log ( b ) , γ = log ( c ) , and the logarithm is taken to be the principal branch.

Proof

Applying the Cauchy integral formula to (45),

(49) ℋ n α ( z ; u ; a , b , c , λ ) = n ! 2 π i ∫ C 1 u α ( a w − u ) α ( e δ + w β − 1 ) α e z w γ d w w n + 1 ,

where C is a circle 0 with radius lesser than ∣ 2 π i − δ β ∣ and δ = log λ u , β = log ( b ) , and γ = log ( c ) are logarithms taken to be the principal branch.

With 2 λ b w u 1 2 sinh δ + w β 2 α = ( e δ + w β − 1 ) α , (49) becomes

(50) ℋ n α ( z ; u ; a , b , c , λ ) = n ! 2 π i ∫ C 1 ( 2 λ u ) α ( a w − u ) α ( b ) α w 2 sinh α δ + w β 2 e z w γ d w w n + 1 .

The shifting of the variable z → z γ + α 2 from (50) gives

(51) ℋ n α z γ + α 2 ; u ; a , b , c , λ = n ! 2 π i 1 ( 2 λ u ) α ∫ C f ( w ) e z w d w w n + 1 ,

where

(52) f ( w ) = c b α w ( a w − u ) α sinh α δ + w β 2

is a meromorphic function with simple poles of order α at the zeros of sinh α δ + w β 2 , which are given by w j = 2 j π i − δ β , j = ± 1 , ± 2 , … .

Taking z → n z and letting n z → ∞ with fixed z ,

(53) ℋ n α n z γ + α 2 ; u ; a , b , c , λ = n ! 2 π i 1 ( 2 λ u ) α ∫ C f ( w ) e n ( z w − log w ) d w w .

It was observed using the saddle point method in Theorem 2.1 that the main contribution of the integrand in (53) originates at the saddle point of the argument of the exponential. Thus, expanding f ( w ) around the saddle point w = z − 1 gives the approximations of ℋ n α n z γ + α 2 ; u ; a , b , c , λ . Consequently, it follows from Lemma 1, Lemma 2, and Theorem 1 of [31] that

(54) ℋ n α n z γ + α 2 ; u ; a , b , c , λ = ( n z ) n ( 2 λ u ) α ∑ k = 0 ∞ f ( k ) ( z − 1 ) k ! p k ( n ) ( n z ) k ,

where p k ( n ) are the polynomials in (11). Solving for the derivatives f ( k ) ( z − 1 ) for k = 0 , 1 , 2 yields

f ( 0 ) ( z − 1 ) = f ( w ) = c b α 2 z ( a 1 ⁄ z − u ) α sinh α z δ + β 2 z f ( 1 ) ( z − 1 ) = c b α 2 z ( a 1 ⁄ z − u ) α sinh α z δ + β 2 z log c b 2 + a 1 ⁄ z log ( a ) a 1 ⁄ z − u − β 2 coth z δ + β 2 z f ( 2 ) ( z − 1 ) = α c b α 2 z ( a 1 ⁄ z − u ) α sinh α z δ + β 2 z log ( a ) a 1 ⁄ z ( a 1 ⁄ z − u ) log ( a ) + α log c b − α β coth z δ + β 2 z + α β 2 coth z δ + β 2 z β 2 − log c b + 1 4 β 2 csch 2 z δ + β 2 z + α log 2 c b .

Expanding the sum in (54) and keeping only the first three terms give (Figure 6)

□ ℋ n α n z γ + α 2 ; u ; a , b , c , λ = ( n z ) n ( 2 λ u ) α f ( 0 ) ( z − 1 ) 0 ! + f ( 1 ) ( z − 1 ) 1 ! p 1 ( n ) n z + f ( 2 ) ( z − 1 ) 2 ! p 2 ( n ) ( n z ) 2 + O 1 n 2 = ( n z ) n c b α 2 z ( a 1 ⁄ z − u ) α ( 2 λ u ) α sinh α z δ + β 2 z 1 − α 2 n z 2 log ( a ) a 1 ⁄ z ( a 1 ⁄ z − u ) log ( a ) + α log c b − α β coth z δ + β 2 z + α β 2 coth z δ + β 2 z β 2 − log c b + 1 4 β 2 csch 2 z δ + β 2 z + α log 2 c b + O 1 n 2 .

Figure 6 Solid lines represent the generalized Apostol-type Frobenius-Euler polynomials of order α \alpha , ℋ n α n z γ + α 2 ; u ; a , b , c , λ {{\mathcal{ {\mathcal H} }}}_{n}^{\alpha }\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{nz}{\gamma }+\frac{\alpha }{2};\hspace{0.33em}u;\hspace{0.33em}a,b,c,\lambda \right) for several values of n n , whereas dashed lines represent the right-hand side of (48) with z ≡ x z\equiv x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 {\left(1+{| \frac{x}{\sigma }| }^{n})}^{-1} where we choose σ = 0.5 \sigma =0.5 . (a) n = 5 , α = 3 , u = 2 , a = 3 , b = 4 , c = 4 , λ = 5 n=5,\alpha =3,u=2,a=3,b=4,c=4,\lambda =5 and (b) n = 6 , α = 4 , u = 3 , a = 5 , b = 4 , c = 3 , λ = 8 n=6,\alpha =4,u=3,a=5,b=4,c=3,\lambda =8 .
Figure 6

Solid lines represent the generalized Apostol-type Frobenius-Euler polynomials of order α , ℋ n α n z γ + α 2 ; u ; a , b , c , λ for several values of n , whereas dashed lines represent the right-hand side of (48) with z ≡ x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 where we choose σ = 0.5 . (a) n = 5 , α = 3 , u = 2 , a = 3 , b = 4 , c = 4 , λ = 5 and (b) n = 6 , α = 4 , u = 3 , a = 5 , b = 4 , c = 3 , λ = 8 .

Remark 4.2

Taking a = 1 , b = e , and c = e , Theorem 4.1 gives a uniform approximation which is similar to that obtained in Theorem 2.1 for the Apostol Frobenius-Euler polynomials of order α .

The following corollaries give the uniform approximations for the generalized Frobenius-Euler polynomials and generalized Euler polynomials of order α with parameter m .

Corollary 4.3

For n , α ∈ Z + , m ∈ R + , u ∈ C \ { 0 , 1 } , and z ∈ C \ { 0 } such that ∣ Im z − 1 ∣ < 2 π + Arg ( u ) m or ∣ z − 1 ∣ < ∣ z − 1 − ( 2 π i + ν ) m ∣ , the generalized Frobenius-Euler polynomials of order α with parameter m satisfy

(55) ℋ n α n z + α 2 ; u ; m = ( n z ) n ( e 1 − m ) α 2 z ( 1 − u ) α ( 2 u ) α sinh α m − z ν 2 z 1 − α 8 n z 2 α m coth m − z ν 2 z m coth m − z ν 2 z − 2 ( 1 − m ) ) + m 2 csch 2 m − z ν 2 z + α ( 1 − m ) 2 + O 1 n 2 ,

where ν = log ( u ) and the logarithm is taken to be the principal branch.

Proof

This follows from Theorem 4.1 by taking a = 1 , b = e m , c = e , and λ = 1 .□

Corollary 4.4

For n , α ∈ Z + , m ∈ R + , and z ∈ C \ 0 such that ∣ Im z − 1 ∣ < π m or ∣ z − 1 ∣ < ∣ z − 1 − π m ∣ , the generalized Euler polynomials of order α with parameter m satisfy

(56) E n α n z + α 2 ; m = ( n z ) n ( e 1 − m ) α 2 z sech m 2 z 1 − α 8 n z 2 α m tanh m 2 z m tanh m 2 z − 2 ( 1 − m ) − m 2 sech 2 m 2 z + α ( 1 − m ) 2 + O 1 n 2 .

Proof

This follows from Theorem 4.1 by taking a = 1 , b = e m , c = e , u = − 1 , and λ = 1 .□

The graphs in Figures 7 and 8 show the approximations of Corollaries 4.3 and 4.4

Figure 7 Solid lines represent the generalized Frobenius-Euler polynomials of order α \alpha with parameter m m , ℋ n α n z + α 2 ; u ; m {{\mathcal{ {\mathcal H} }}}_{n}^{\alpha }\left(\phantom{\rule[-0.75em]{}{0ex}},nz+\frac{\alpha }{2};\hspace{0.33em}u;\hspace{0.33em}m\right) for several values of n n , whereas dashed lines represent the right-hand side of (55) with z ≡ x z\equiv x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 {\left(1+{| \frac{x}{\sigma }| }^{n})}^{-1} where we choose σ = 0.5 \sigma =0.5 . (a) n = 4 , α = 5 , m = 6 , u = 5 n=4,\alpha =5,m=6,u=5 and (b) n = 7 , α = 4 , m = 3 , u = 6 n=7,\alpha =4,m=3,u=6 .
Figure 7

Solid lines represent the generalized Frobenius-Euler polynomials of order α with parameter m , ℋ n α n z + α 2 ; u ; m for several values of n , whereas dashed lines represent the right-hand side of (55) with z ≡ x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 where we choose σ = 0.5 . (a) n = 4 , α = 5 , m = 6 , u = 5 and (b) n = 7 , α = 4 , m = 3 , u = 6 .

Figure 8 Solid lines represent the generalized Euler polynomials of order α \alpha with parameter m m , ℰ n α n z + α 2 ; m {{\mathcal{ {\mathcal E} }}}_{n}^{\alpha }\left(\phantom{\rule[-0.75em]{}{0ex}},nz+\frac{\alpha }{2};\hspace{0.33em}m\right) for several values of n n , whereas dashed lines represent the right-hand side of (56) with z ≡ x z\equiv x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 {\left(1+{| \frac{x}{\sigma }| }^{n})}^{-1} where we choose σ = 0.5 \sigma =0.5 . (a) n = 4 , α = 5 , m = 6 , u = 5 n=4,\alpha =5,m=6,u=5 and (b) n = 7 , α = 4 , m = 3 , u = 6 n=7,\alpha =4,m=3,u=6 .
Figure 8

Solid lines represent the generalized Euler polynomials of order α with parameter m , ℰ n α n z + α 2 ; m for several values of n , whereas dashed lines represent the right-hand side of (56) with z ≡ x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 where we choose σ = 0.5 . (a) n = 4 , α = 5 , m = 6 , u = 5 and (b) n = 7 , α = 4 , m = 3 , u = 6 .

The following table shows the exact and approximate values of the generalized Apostol-type Frobenius-Euler polynomials of order α , ℋ n α n z γ + α 2 ; u , a , b , c ; λ , where n = 4 , α = 3 , u = 2 , a = 2 , b = 3 , c = 4 , λ = 5 , and γ = log c = log 4 for some increasing values of z = x .

It can also be observed from Table 2 that for increasing values of z , the relative error decreases.

Table 2

Exact and approximate values of ℋ 4 3 4 z log 4 + 3 2 ; 2 ; 2 , 3 , 4 ; 5

z Exact values ℋ 4 3 4 z log 4 + 3 2 ; 2 , 3 , 4 ; 5 Approximate values using Theorem 4.1 Relative error
3 ‒ 6.22627 × 10 ‒ 8.05204 × 10 2.93238 × 10
4 ‒ 4.54829 × 1 0 2 ‒ 4.99541 × 1 0 2 9.83059 × 1 0 ‒ 2
5 ‒ 1.6595 × 1 0 3 ‒ 1.74272 × 1 0 3 5.01423 × 1 0 ‒ 2
6 ‒ 4.3878 × 1 0 3 ‒ 4.52054 × 1 0 3 3.02532 × 1 0 ‒ 2
7 ‒ 9.57876 × 1 0 3 ‒ 9.77113 × 1 0 3 2.00824 × 1 0 ‒ 2
8 ‒ 1.8399 × 1 0 4 ‒ 1.86604 × 1 0 4 1.42037 × 1 0 ‒ 2
9 ‒ 3.22427 × 1 0 4 ‒ 3.25819 × 1 0 4 1.05171 × 1 0 ‒ 2
10 ‒ 5.27317 × 1 0 4 ‒ 5.31569 × 1 0 4 8.06453 × 1 0 ‒ 3
11 ‒ 8.17151 × 1 0 4 ‒ 8.22346 × 1 0 4 6.35733 × 1 0 ‒ 3
12 ‒ 1.2127 × 1 0 5 ‒ 1.21891 × 1 0 5 5.12551 × 1 0 ‒ 3

4.2 Enlarged region of validity

In Section 3, approximation of the Apostol-Frobenius-Euler polynomials of order α is obtained using the method of contour integration that introduced the incomplete gamma function in the formula. Using the same technique, approximation for the generalized Apostol-type Frobenius-Euler polynomials of order α with parameters a , b , and c is given in the following theorem.

Theorem 4.5

For n , α ∈ Z + , a , b , and c ∈ R + , u , λ ∈ C \ { 0 , 1 } , and z ∈ C such that ∣ z − 1 ∣ < ∣ z − 1 − w k ∣ for all k = l + 1 , l + 2 , … , the generalized Apostol-type Frobenius-Euler polynomials of order α satisfy

(57) ℋ n α n z γ + α 2 ; u ; a , b , c , λ = 1 ( 2 λ u ) α ∑ k = 1 l ∑ j = 1 α e w k n z r k j ∑ s = 0 n n s ( − 1 ) ( j − 1 ) ⟨ j − 1 ⟩ s ( w k ) − ( j − 1 + s ) × ( n − s ) ! w k n − s + 1 − Γ ( n − s + 1 , w k n z ) w k n − s + 1 + ( − 1 ) j ⟨ j ⟩ n w k j + n + ( n z ) n ∑ k = 0 ∞ f ( k ) ( z − 1 ) − h l ( k ) ( z − 1 ) k ! p k ( n ) ( n z ) k ,

where the polynomials p k ( n ) are given in (11) and h l k is the kth derivative of the function h l ( w ) given by (28) and

∑ j = 1 m r k j ( w − w k ) j ,

are the given principal parts of the Laurent series corresponding to the poles w k = ( 2 k π i − δ ) ⁄ β , where δ = log λ u and β = log ( b ) and the entire function h ( w ) is determined by f ( w ) = c b α w ( a w − u ) α sinh α δ + w β 2 .

Proof

Recall from (53),

(58) ℋ n α n z γ + α 2 ; u ; a , b , c , λ = n ! 2 π i 1 ( 2 λ u ) α ∫ C f ( w ) e n ( z w − log w ) d w w ,

where

f ( w ) = c b α w ( a w − u ) α sinh α δ + w β 2 ,

with δ = log λ u and β = log ( b ) . Substituting (16) into (58) gives

(59) ℋ n α n z γ + α 2 ; u ; a , b , c , λ = S l n , a ( z ) + T l n , a ( z ) ,

where

(60) S l n , α ( z ) = n ! 2 π i 1 ( 2 λ u ) α ∫ C f l ( w ) e w n z d w w n + 1 ,

(61) T l n , α ( z ) = 1 ( 2 λ u ) α ∫ C n ! 2 π i ∑ k = 1 l ∑ j = 1 α r k j ( w − w k ) j e w n z d w w n + 1 .

Repeating the process used in Theorem 3.1 to expand f l ( w ) around the saddle point z − 1 gives the expansion of S l n , α ( z ) as

(62) S l n , α ( z ) = ( n z ) n 1 ( 2 λ u ) α ∑ k = 0 ∞ f l ( k ) ( z − 1 ) − h l ( k ) ( z − 1 ) k ! p k ( n ) ( n z ) k

valid for ∣ z − 1 ∣ < ∣ z − 1 − w j ∣ , j = l + 1 , l + 2 , … and z ≠ 0 .

Conversely, the expansion of T l n , α can be obtained by employing the same method to evaluate the integral of Y l n , α in (21). The method of contour integration involves shifting the integration contour by w = w k + t , which resulted in an expansion with incomplete gamma function. Thus, the expansion is given as

(63) T l n , α ( z ) = 1 ( 2 λ u ) α ∑ k = 1 l ∑ j = 1 α e w k n z r k j ∑ s = 0 n n s ( − 1 ) ( j − 1 ) ⟨ j − 1 ⟩ s ( w k ) − ( j − 1 + s ) × ( n − s ) ! w k n − s + 1 − Γ ( n − s + 1 , w k n z ) w k n − s + 1 + ( − 1 ) j ⟨ j ⟩ n w k j + n .

Substituting (62) and (63) into (59) gives

(64) ℋ n α n z γ + α 2 ; u ; a , b , c , λ = 1 ( 2 λ u ) α ∑ k = 1 l ∑ j = 1 α e w k n z r k j ∑ s = 0 n n s ( − 1 ) ( j − 1 ) ⟨ j − 1 ⟩ s ( w k ) − ( j − 1 + s ) × ( n − s ) ! w k n − s + 1 − Γ ( n − s + 1 , w k n z ) w k n − s + 1 + ( − 1 ) j ⟨ j ⟩ n w k j + n + ( n z ) n ∑ k = 0 ∞ f ( k ) ( z − 1 ) − h l ( k ) ( z − 1 ) k ! p k ( n ) ( n z ) k ,

where the polynomials p k ( n ) are given in (11) and h l k is the k th derivative of the function h l ( w ) given by (28) and

∑ j = 1 m r k j ( w − w k ) j

are the given principal parts of the Laurent series corresponding to the poles w k = ( 2 k π i − δ ) ⁄ β , where δ = log λ u and β = log ( b ) and the entire function h ( w ) is determined by f ( w ) = c b α w ( a w − u ) α sinh α δ + w β 2 .□

The comparison of the accuracy of the asymptotic formula obtained in (48) and (57) is shown in Figure 9.

Figure 9 Solid lines in (a) and (b) represent ℋ n α n z γ + α 2 ; u ; a , b , c , λ {{\mathcal{ {\mathcal H} }}}_{n}^{\alpha }\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{nz}{\gamma }+\frac{\alpha }{2};\hspace{0.33em}u;\hspace{0.33em}a,b,c,\lambda \right) whereas dashed lines in (a) and (b) represent the right-hand side of (6) and (15) for n = 2 , α = 2 , u = 3 , a = 5 , b = 3 , c = 2 n=2,\alpha =2,u=3,a=5,b=3,c=2 , and λ = 2 \lambda =2 , respectively, with z ≡ x z\equiv x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 {\left(1+{| \frac{x}{\sigma }| }^{n})}^{-1} where we choose σ = 0.5 \sigma =0.5 . (a) n = 3 , α = 3 , u = 4 , λ = 6 n=3,\alpha =3,u=4,\lambda =6 and (b) n = 3 , α = 3 , u = 4 , λ = 6 n=3,\alpha =3,u=4,\lambda =6 .
Figure 9

Solid lines in (a) and (b) represent ℋ n α n z γ + α 2 ; u ; a , b , c , λ whereas dashed lines in (a) and (b) represent the right-hand side of (6) and (15) for n = 2 , α = 2 , u = 3 , a = 5 , b = 3 , c = 2 , and λ = 2 , respectively, with z ≡ x , both normalized by the factor ( 1 + ∣ x σ ∣ n ) − 1 where we choose σ = 0.5 . (a) n = 3 , α = 3 , u = 4 , λ = 6 and (b) n = 3 , α = 3 , u = 4 , λ = 6 .

5 Conclusion

The study of special functions, particularly the investigation of Apostol-type polynomials, has proven to be a captivating area in mathematics. These special functions exhibit unique properties and have a wide range of applications in various fields of knowledge. The development of topics in special functions involves exploring combinations, extensions, and generalizations of classical polynomials such as Bernoulli, Euler, Genocchi, and tangent polynomials.

This article focused on the Apostol-Frobenius-Euler polynomials of order a , denoted by H n α ( z ; u ; λ ) , and their higher-order approximations. Building upon the saddle point method, Corcino et al. [27] derived approximations for the higher-order tangent polynomials and introduced a new method to extend the validity region of these approximations. The findings of this study indicate that these methods can also be applied to the higher-order Apostol-Frobenius-Euler polynomials.

Consequently, the article obtained approximations of the higher-order Apostol-Frobenius-Euler polynomials using hyperbolic functions, specifically for large values of the parameter n . Additionally, uniform approximations with an enlarged region of validity were derived. Furthermore, the same methods were applied to obtain approximations of the generalized Apostol-type Frobenius-Euler polynomials of order α with parameters a , b , and c .

To assess the accuracy of the exact values and the corresponding approximations of these polynomials, the article presented tables of values and graphs for specific values of the parameters. These tables and graphs provide visual evidence of the effectiveness of the derived approximations.

In summary, the research presented in this article contributes to the field of special functions by expanding our understanding of Apostol-type polynomials, specifically the Apostol-Frobenius-Euler polynomials. The developed methods for obtaining higher-order approximations and enlarging the validity region demonstrate their applicability and effectiveness. The obtained results and graphical illustrations highlight the accuracy and usefulness of these approximations, providing valuable insights for further research and practical applications in various disciplines.

For future research work, it is recommended to explore the applicability of the methods used in this article to Apostol-Frobenius-Genocchi polynomials of the higher order.

Acknowledgement

The authors would like to thank the referees for reviewing the paper thoroughly and to Cebu Normal University for funding this research through its Research Institute for Computational Mathematics and Physics (RICMP).

  1. Funding information: This research was funded by Cebu Normal University through its Research Institute for Computational Mathematics and Physics (Grant Number CNU RICMP Project 4-2022).

  2. Author contributions: Conceptualization (C. Corcino, W.D. Castañeda, Jr. and R. Corcino), formal analysis (C. Corcino, W.D. Castañeda, Jr., and R. Corcino), funding acquisition (C. Corcino and R. Corcino), investigation (C. Corcino, W.D. Castañeda, Jr., and R. Corcino), methodology (C. Corcino, W.D. Castañeda, Jr. and R. Corcino), supervision (C. Corcino and R. Corcino), writing – original draft (W.D. Castañeda, Jr.), writing – review and editing (C. Corcino and R. Corcino). All authors have read and agreed to the published version of the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Ethical approval: The conducted research is not related to either human or animal use.

  5. Data availability statement: The newly added information in this study are available on request from the corresponding author.

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Received: 2023-01-05
Revised: 2023-06-01
Accepted: 2023-07-11
Published Online: 2024-08-24

© 2024 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/dema-2023-0106
Keywords for this article
Apostol-Frobenius-Euler polynomials; generalized Apostol-type Froebnius-Euler polynomials; Euler Polynomials; Hermite polynomials; asymptotic approximation
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  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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