N.Alam,W.A.Khan,S.Obeidat,G.Muhiuddin,N.S.Diab,H.N.Zaidi,A.Altaleb and L.Bachioua
1Department of Basic Sciences,Deanship of Preparatory Year,University of Ha’il,Ha’il,2440,Saudi Arabia
2Department of Mathematics and Natural Sciences,Prince Mohammad Bin Fahd University,Al Khobar,31952,Saudi Arabia
3Department of Mathematics,Faculty of Science,University of Tabuk,Tabuk,71491,Saudi Arabia
ABSTRACT In this article, we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials.Some fundamental properties of these functions are given.By using these generating functions and some identities,relations among trigonometric functions and two parametric kinds of Bell-based Bernoulli and Euler polynomials,Stirling numbers are presented.Computational formulae for these polynomials are obtained.Applying a partial derivative operator to these generating functions,some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained.In addition,some remarks and observations on these polynomials are given.
KEYWORDS Bernoulli polynomials;euler polynomials;bell polynomials;stirling numbers
Special polynomials and numbers possess much importance in multifarious areas of sciences such as physics,mathematics,applied sciences,engineering and other related research fields covering differential equations,number theory,functional analysis,quantum mechanics,mathematical analysis,mathematical physics and so on (see [1–22]) and see also each of the references cited therein.For example, Bernoulli polynomials and numbers are closely related to the Riemann zeta function,which possesses a connection with the distribution of prime numbers.Some of the most significant polynomials in the theory of special polynomials are the Bell,Euler,Bernoulli,Hermite,and Genocchi polynomials.Recently,the aforesaid polynomials and their diverse generalizations have been densely considered and investigated by many physicists and mathematicians (see [1–18,22]) and see also the references cited therein (see [6–9,14–17]).The class of Appell polynomial sequence is one of the significant classes of polynomials sequence [1].In applied mathematics, theoretical physics,approximation theory,and several other mathematics branches.The set of Appell polynomial sequence is closed under the operation of umbral composition of polynomial sequences.The Appell polynomial sequence can be given by the following generating function:
The power seriesA(z)given by
whereAi{i=1,2,3,···}are real coffiecients.It is easy to see that for anyA(z),the derivative ofA(z)satisfies
The Bell-based Bernoulli and Bell-based Bernoulli polynomials of the first kind are the special cases of Appell polynomials(see[2,18]).
The generalized Bernoulli and Euler polynomials of orderαare defined by(see[2–5])
and
respectively.
At the pointξ= 0,=(0)andare the Bernoulli and Euler numbers of orderα.
Forj≥0,Stirling numbers of the first kind are defined by
where(ξ)0=1,and(ξ)j=ξ(ξ-1)···(ξ-j+1),(ξ≥1).From(4),we see that
The Stirling numbers of the second kind are defined by
By(5),we note that
The generating function of Bell polynomialsBelj(ξ)are defined by(see[7])
In the special caseξ= 1,Belj=Belj(1),(j≥0)are the Bell numbers.From (1.7) and (1.8),we have
Recently, Duran et al.[2] introduced the generalized Bell-based Bernoulli polynomials are defined by
At the pointξ=η=1,Bel=Bel(1;1)are the generalized Bell-based Bernoulli numbers.
Kim et al.[13]and Jamei et al.[4,5]introduced the Bernoulli and Euler polynomials of complex variable are defined by
and
respectively.
Also they have prove that(see[4,5,8,9,19,20,21])
and
where
and
Motivated by the importance and potential applications in certain problems in number theory,combinatorics, classical and numerical analysis and physics, several families of Bernoulli and Euler polynomials and special polynomials have been recently studied by many authors, see [8,9,19–21].Recently, Kim et al.[13,16] have introduced the degenerate Bernoulli and degenerate Euler polynomials of a complex variable.By separating the real and imaginary parts, they introduced the parametric kinds of these degenerate polynomials.The manuscript of this paper is arranged as follows.In Section 2, we introduce parametric kinds of Bell-based Bernoulli polynomials and prove several identities of Bell-based Bernoulli polynomials by using different analytical means and applying generating functions.In Section 3,we establish parametric kinds of Bell-based Euler polynomials and investigate some identities of these polynomials.
In this section,we consider the Bell-based Bernoulli polynomials of complex variable and deduce some identities of these polynomials.First,we present the following definition as
On the other hand,we suppose that
Thus,by(19)and(20),we have
and
From(21)and(22),we get
and
respectively.
Note thatBel(ξ,0;0)=(ξ),0,0;0)=0,(j≥0).
From(23)–(26),we have
Remark 2.1.Forξ=ζ=0 in(25)and(26),we get
and
respectively.
It is clear that
Remark 2.2.Lettingζ=0 in(25)and(26),we obtain
and
respectively.
Remark 2.3.On settingξ=0 in(25)and(26),we acquire
and
respectively.
Now,we start some basic properties of these polynomials.
Theorem 2.1.Letj≥0.Then
and
Proof.By(33)and(34),we can derive the following equations:
and
Therefore,by(37)and(38),we get(35).Similarly,we can easily obtain(36).
Theorem 2.2.Letj≥0.Then
and
Proof.By using(21)and(22),we obtain(39)and(40).So we omit the proof.
Theorem 2.3.Letj≥0.Then
and
Proof.Consider
Now
which proves(41).The proof of(42)is similar.
Theorem 2.4.For everyj∈Z+,we have
Proof.Using(25)and(26),we obtain(43)–(46).Here,we omit the proof of the theorem.
Theorem 2.5.Letj≥0.Then
and
Proof.By changingξwithξ+sin(25),we have
which complete the proof(47).The result(48)can be similarly proved.
Theorem 2.6.Letj≥1.Then
and
Proof.Eq.(25)yields
proving(49).Other(50),(51)and(52)can be similarly derived.
Theorem 2.7.Letj≥1.Then
and
Proof.By(25),we have
The complete proof of(53).The proof of(54)is similar.
Theorem 2.8.Forj≥0.Then
and
Proof.By(25),we have
On the other hand,we have
In view of(58)and(59),we get(56).Similarly,we can easily obtain(57).
Theorem 2.9.Letj≥0.Then
and
Proof.In definition 2.1,we have
On the other hand,we have
Therefore,by(62)and(63),we obtain(60).Similarly,we can easily obtain(61).
Theorem 2.10.Letj≥0.Then
and
Proof.Using(7)and(25),we find
In view of(25)and(66),we get(64).Similarly,we can easily obtain(65).
In this section, we define Bell-based Euler polynomials of complex variable and derive some explicit expressions of these polynomials.Now we start with the following definition as
By using(67)and(20),we have
and
From(68)and(69),we get
and
Definition 3.1.Letj≥0.We define two parametric kinds of cosine Bell-based Euler polynomialsBel(ξ,η;ζ)and sine Bell-based Euler polynomialsBel(ξ,η;ζ), for non negative integerjare defined by
and
respectively.
From(70)–(73),we have
Note that
Remark 3.1.Forξ=0 in(72)and(73),we get
and
respectively.
Remark 3.2.Lettingζ=0 in(72)and(73),we obtain
and
respectively.
Remark 3.3.On takingξ=ζ=0 in(72)and(73),we acquire
and
respectively.
Theorem 3.1.Letj≥0.Then
and
Proof.From(78)and(79),we can derive the following equations:
and
Therefore,by(82)and(83),we get(80).Similarly,we can easily obtain(81).
Theorem 3.2.Letj≥0.Then
and
Proof.By using(68)and(69),we can easily get(84)and(85).So we omit the proof.
Theorem 3.3.Letj≥0.Then
and
Proof.Consider the identity,we have
Now
which proves(86).The proof of(87)is similar.
Theorem 3.4.Letj≥0.Then
Proof.Using(72)and(73),we obtain(88)–(90).Here,we omit the proof of the theorem.
Theorem 3.5.Letj≥0.Then
and
Proof.By changingξwithξ+sin(72),we have
which proves(91).The result(92)can be similarly proved.
Theorem 3.6.Letj≥1.Then
and
Proof.Eq.(72)yields
proving(93).Other(94)–(96)can be similarly derived.
Theorem 3.7.Letj≥0.Then
and
Proof.By definition(72),we have
The complete proof of the result(97).The proof of(98)is similar.
Theorem 3.8.Letj≥0.Then
Proof.Using definition 3.1,we have
On the other hand,we have
In view of(102)and(103),we get(100).Similarly,we can easily obtain(101).
Theorem 3.9.Letj≥0.Then
and
Proof.Using(7)and(72),we find
In view of(72)and(106),we get(104).Similarly,we can easily obtain(105).
In this section, certain zeros of the Bell-based Bernoulli polynomials of complex variable,η;ζ)and,η;ζ)and beautifully graphical representations are shown.
The first few of them are
Table 1 shows some numerical values of Bell-based Bernoulli polynomials of a complex variable.
Table 1:Numerical values of Bell-based Bernoulli polynomials of a complex variable Bel(ξ,η;ζ)and Bel(ξ,η;ζ)
Table 1:Numerical values of Bell-based Bernoulli polynomials of a complex variable Bel(ξ,η;ζ)and Bel(ξ,η;ζ)
j BelB(α,c)j (5,6;2) BelB(α,s)j (5,6;2)0 1.0 0 1 6.0 6.0 2 1.83333 72.0 3-397.0 465.0 4-5123.9 840.0 5-38622.3 -21381.0 6-158258 -341508 7 505053 318760×101 8 1.81135×107 -2.1281×107 9 2.29789×108 -8.71221×108 10 2.19881×109 2.12412×108
Fig.1 shows the plot for the Bell-based Bernoulli polynomialBel(ξ,η;ζ)with(ξ,η;ζ)=(5,6;2).Fig.2 shows the plot for 3D Bell-based Bernoulli polynomialsBel(ξ,η;ζ).
Figure 1:Bell-based Bernoulli polynomials Bel(ξ,η;ζ)
Figure 2:3D Bell-based Bernoulli polynomials Bel(ξ,η;ζ)
In this section, certain zeros of the Bell-based Bernoulli polynomials of complex variableBel(ξ,η;ζ)andBel(ξ,η;ζ)and beautifully graphical representations are shown.
The first few of them are
Table 2 shows some numerical values of Numerical values of Bell-based Euler polynomials of complex variable.
Table 2:Numerical values of Bell-based Euler polynomials of a complex variable (ξ,η;ζ)and(ξ,η;ζ)
Table 2:Numerical values of Bell-based Euler polynomials of a complex variable (ξ,η;ζ)and(ξ,η;ζ)
j BelE(α,c)j (5,6;2) BelE(α,s)j (5,6;2)0 1.0 0 1 6.0 6.0 2 1.5 72.0 3-403.0 459.0 4-5127.0 696.0 5-37282.0 -22914.0 6-132625 -345096 7 767443 -302880×101 8 1.93872×107 -1.80647×107 9 2.21045×108 -5.04831×107 10 1.91065×109 4.90648×108
Fig.3 shows the plot for the Bell-based Euler polynomial,η;ζ)with(ξ,η;ζ)=(5,6;2).
Figure 3:Bell-based Euler polynomials (ξ,η;ζ)
In the present article,we have considered the parametric kinds of Bell-based Bernoulli and Euler polynomials by making use of the exponential as well as trigonometric functions.We have also derived some analytical properties of our newly introduced parametric polynomials by using the series manipulation technique.Furthermore,it is noticed that,if we consider any Appell polynomials of a complex variable(as discussed in the present article),then we can easily define its parametric kinds by separating the complex variable into real and imaginary parts.Consequently,the results of this article may potentially be used in mathematics,mathematical physics and engineering.
Acknowledgement:The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.
Funding Statement:This research was funded by Research Deanship at the University of Ha’il,Saudi Arabia,through Project No.RG-21 144.
Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.
Computer Modeling In Engineering&Sciences2023年4期