It is well-known that the Chebyshev polynomials play vital roles in the scope of mathematical analysis and its applications. The first- and second-kinds are special symmetric polynomials of the Jacobi polynomials, so they are ultraspherical polynomials. These two kinds of Chebyshev polynomials are the most popular kinds, and they are employed extensively in numerical analysis, see for example [1–4]. There are other four kinds of Chebyshev polynomials. Third- and fourth-kinds are also special kinds of Jacobi polynomials, but they differ from the first- and second-kinds since they are not ultraspherical polynomials. In fact, they are non-symmetric Jacobi polynomials with certain parameters. There are considerable publications devoted to investigating these kinds of polynomials from both theoretical and practical points of view. For some articles in this direction, one can be referred to [5–7]. Fifth- and sixth-kinds of Chebyshev polynomials are not special cases of the Jacobi polynomials. These polynomials were first established by Jamei in his Ph.D. dissertation [8]. Recently, these kinds of polynomials have been employed for treating numerically some types of differential equations, see [9–11].

Spectral methods are essential in numerical analysis. These methods have been utilized fruitfully for obtaining numerical solutions of various kinds of differential and integral equations. The philosophy in the application of spectral methods is based on writing the desired numerical solution in terms of certain “basis functions,” which may be expressed as combinations of various orthogonal polynomials. There are three celebrated spectral methods; they are tau, collocation, and Galerkin methods. Galerkin method is based on choosing suitable combinations of orthogonal polynomials satisfying the underlying boundary/initial conditions [12]. Tau method is more flexible than the Galerkin method due to the freedom in choosing the basis functions. Collocation method is the most popular method because of its capability to treat all kinds of differential equations. There are considerable contributions employ this method in different types of differential equations [13–15]. For a survey on spectral methods and their applications, one can consult [16–21].

The problems of establishing derivatives and integrals formulas of different orthogonal polynomials in terms of their original polynomials are of great interest. These formulas play important parts in obtaining spectral solutions of various differential equations. For example, the authors in [5,6] derived explicit formulas of the high-order derivatives and repeated integrals of Chebyshev polynomials of third- and fourth-kinds, and after that, they utilized the derived formulas for obtaining spectral solutions of some boundary value problems (BVPs).

Boundary value problems have important roles due to their appearance in almost all branches related to applied sciences such as engineering, fluid mechanics, and optimization theory (see, [22]). Some examples concerning applications of BVPs are given with details in [22]. Moreover, the author showed that, at some times, it is more appropriate to transform high-order BVPs into systems of differential equations. Many real-life phenomena can be described by high-order BVPs.

Regarding the even-order BVPs, they arise in numerous problems. The free vibrations analysis of beam structures is governed by a fourth-order ordinary differential equation [23]. The vibration behavior of a ring structures is governed by a sixth-order ordinary differential equation [24]. Eighth-order BVPs arise in torsional vibration of uniform beams [25]. In addition, 10th- and 12th-orders appear in applications. When a uniform magnetic field is applied across the fluid in the same direction as gravity. Ordinary convection and overstability yield a 10th-order and a 12th-order BVPs, respectively [26]. For applications regarding the high-order BVPs appear in hydrodynamic and hydromagnetic stability, fluid dynamics, astronomy, beam and long wave theory [27,28]. For some other applications of these kinds of problems, one can be referred to [29–31].

Several studies were performed for the numerical solutions of these types of equations. Some of the algorithms treat these kinds are the perturbation and homotopy perturbation methods in [32], differential transform methods in [26] and spectral methods in [15,33–37].

The main objectives of this article can be summarized in the following items:

Deriving high-order derivatives formulas of certain classes of polynomials that generalize Chebyshev polynomials of the third- and fourth-kinds.

The current paper is organized as follows. In Section 2, we give some relevant properties of Jacobi polynomials. In Section 3, we derive in detail two new formulas which give explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one. In Section 4, we present a spectral Galerkin algorithm for solving some high even-order linear BVPs with constant coefficients. Section 5 is devoted to presenting another algorithm for treating linear and nonlinear BVPs based on the application of a suitable collocation algorithm. Section 6 is interested in solving the connection problem between certain Jacobi polynomials and the Legendre polynomials. In Section 7, we investigate the convergence and error analysis of the proposed expansion in Section 5. In Section 8, we present some numerical examples including comparisons with some other methods to ascertain the accuracy and efficiency of the proposed algorithms presented in Sections 4 and 5. Finally, some conclusions are reported in Section 9.

The classical Jacobi polynomials Pm(α,β)(x)(m=0,1,2,…), is a polynomial of degree m which can be defined in hypergeometric form as Pm(α,β)(x)=Γ(m+α+1)m!Γ(α+1)2F1(-m,m+θ;α+1|1-x2)=Γ(m+β+1)m!Γ(β+1)2F1(-m,m+θ;β+1|1+x2), where θ=α+β+1.

It is easy to see the following special values

(1) Pm(α,β)(-x)=(-1)mPm(β,α)(x),Pm(α,β)(1)=Γ(m+α+1)m!Γ(α+1),Pm(α,β)(-1)=(-1)mΓ(m+β+1)m!Γ(β+1).

It is desirable to define the normalized orthogonal Jacobi polynomials as [38]

(2) Jm(α,β)(x)=m!Γ(α+1)Γ(m+α+1)Pm(α,β)(x),

which gives Jm(α,β)(1)=1(m=0,1,2,…). We have the following six special cases of the normalized Jacobi polynomials. Cm(α)(x)=Jm(α-12,α-12)(x),Tm(x)=Jm(-12,-12)(x),Um(x)=(m+1)Jm(12,12)(x),Vm(x)=Jm(-12,12)(x),Wm(x)=(2m+1)Jm(12,-12)(x),Pm(x)=Jm(0,0)(x), where Cm(α)(x), T_m(x), U_m(x), V_m(x), W_m(x) and P_m(x) are the ultraspherical, Chebyshev of the first-, second-, third- and fourth-kinds, and the Legendre polynomials, respectively.

The polynomials Jm(α,β)(x) may be generated using the following recurrence relation:

(3) 2(m+θ)(m+α+1)(2m+θ-1)Jm+1(α,β)(x)=(2m+θ-1)(2m+θ)(2m+θ+1)xJm(α,β)(x)+(α2-β2)(2m+θ)Jm(α,β)(x)-2m(m+β)(2m+θ+1)Jm-1(α,β)(x),m=1,2,…,

with the initial values: J0(α,β)(x)=1,J1(α,β)(x)=12(α+1)[α-β+(θ+1)x].

The orthogonality relation satisfied by Jn(α,β)(x) is given by:

(4) ∫ -11(1-x)α(1+x)βJm(α,β)(x)Jn(α,β)(x)dx=hnα,βδmn,

where δmn is the well-known Kronecker delta function, and hnα,β=2θn!Γ(n+β+1)[Γ(α+1)]2(2n+θ)Γ(n+θ)Γ(n+α+1).

The special values

(5) DqJm(α,β)(1)=∏ k=0q-1(m-k)(m+k+θ)2(k+α+1),

(6) DqJm(α,β)(-1)=(-1)m+qΓ(α+1)Γ(m+β+1)Γ(β+1)Γ(m+α+1)DqJm(β,α)(1),

will be of important use later.

It is useful to extend the Jacobi polynomials on a general interval [a, b]. In fact, the shifted Jacobi polynomials can be defined on [a, b] by

(7) J̃i(α,β)(x)=Ji(α,β)(2x-a-bb-a).

Hence, properties and relations concerned with the Jacobi polynomials Ji(α,β)(x) can be easily transformed to give their counterparts for the shifted polynomials J̃i(α,β)(x).

For our upcoming computations, we need the orthogonality relation of J̃n(α,α+1)(x) on [a, b]. This relation is given explicitly as:

(8) ∫ ab(b-x)α(x-a)α+1J̃m(α,α+1)(x)J̃n(α,α+1)(x)dx=h̃nαδmn,

where h̃nα is given by:

(9) h̃nα=(b-a)2α+2n![Γ(α+1)]22Γ(n+2α+2).

The main objective of this section is to state and prove two theorems which express the high-order derivatives of the shifted Jacobi polynomials J̃n(α,α+1)(x) and J̃n(α+1,α)(x) in terms of their original polynomials. The basic idea behind the derivation of the desired formulas is built on employing the power form representation and the inversion formula of the Jacobi polynomials Jn(α,α+1)(x). Now, the following three lemmas are essential in this regard.

Lemma 3.1. The explicit power form of the normalized Jacobi polynomials Jn(α,α+1)(x), n≥1, is given by

(10) Jn(α,α+1)(x)=∑k=0[n2]an,kxn-2k+∑k=0[n2]bn,kxn-2k-1,

where

(11) an,k=(-1)kn!Γ(α+1)2n-2k+2α+1Γ(n-k+α+32)πk!(n-2k)!Γ(n+2α+2),

and

(12) bn,k=(-1)k+1n!Γ(α+1)2n-2k+2αΓ(n-k+α+12)πk!(n-2k-1)!Γ(n+2α+2),

where [z] represents the largest integer less than or equal to z.

Proof. We proceed by induction on n. Assume that relation (10) is valid for (n −1) and (n −2), and we have to prove the validity of (10) itself. Starting with the recurrence relation (3), (for the case β=α+1) in the form

(13) Jn(α,α+1)(x)=(αnx+βn)Jn-1(α,α+1)(x)+γnJn-2(α,α+1)(x),

where αn=2n+2α+1n+2α+1,βn=-(2α+1)(n+2α+1)(2n+2α-1),γn=-(n-1)(2n+2α+1)(n+2α+1)(2n+2α-1),

then, the application of the induction hypothesis twice yields

(14) Jn(α,α+1)(x)=αn{∑k=0[n-12]an-1,kxn-2k+∑k=0[n-12]bn-1,kxn-2k-1}+βn{∑k=0[n-12]an-1,kxn-2k-1+∑k=0[n-12]bn-1,kxn-2k-2}+γn{∑k=0[n2]-1an-2,kxn-2k-2+∑k=0[n2]-1bn-2,kxn-2k-3}.

Eq. (14) may be written in the form Jn(α,α+1)(x)=∑1+∑2, where ∑1=(βnbn-1,n2-1+γnan-2,n2-1)δn+αnan-1,0xn+∑k=1[n-12]{αnan-1,k+βnbn-1,k-1+γnan-2,k-1}xn-2k,∑2=(αnbn-1,0+βnan-1,0)xn-1+∑k=1[n-12]{αnbn-1,k+βnan-1,k+γnbn-2,k-1}xn-2k-1, and δn={1,n even, 0,n odd.

It is not difficult to see that αnan-1,k+βnbn-1,k-1+γnan-2,k-1=an,k,1≤k≤[n-12],αnbn-1,k+βnan-1,k+γnbn-2,k-1=bn,k,1≤k≤[n-12],αnan-1,0=an,0,βnbn-1,n2-1+γnan-2,n2-1=an,n2,αnbn-1,0+βnan-1,0=bn,0, where a_{n, k}, b_{n, k} are given by (11) and (12), respectively. Therefore ∑1=∑k=0[n2]an,kxn-2k,

and ∑2=∑k=0[n2]bn,kxn-2k-1, and this completes the proof of formula (10).

Lemma 3.2. For all m∈ℤ+, the following inversion formula holds

(15) xm=∑i=0[m2]ci,mJm-2i(α,α+1)(x)+∑i=0[m-12]di,mJm-2i-1(α,α+1)(x),

where ci,m=πm!2-m-2α-1Γ(m-2i+2α+2)i!Γ(α+1)(m-2i)!Γ(m-i+α+32), and di,m=πm!2-m-2α-1Γ(-2i+m+2α+1)i!Γ(α+1)(m-2i-1)!Γ(m-i+α+32).

Proof. We proceed by induction on m. Form = 1, the left-hand side equals the right-hand side. Assume that relation (15) is valid, and we have to show that

(16) xm+1=∑i=0[m+12]ci,m+1Jm-2i+1(α,α+1)(x)+∑i=0[m2]di,m+1Jm-2i(α,α+1)(x).

Multiplying both sides of relation (15) by x, then with the aid of relation (3), (for the case β=α+1), and after some rather manipulation, Eq. (16) can be proved.

Lemma 3.3. For every nonnegative integer m, we have

(17) ∑j=0m(-1)jΓ(-j+n+α+12)j!(m-j)!Γ(-j+n-q-m+α+32)=(q+m-1)!Γ(n-m+α+12)(q-1)!m!Γ(n-q-m+α+32).

Proof. If we set Sn,q,m=∑j=0m(-1)jΓ(-j+n+α+12)j!(m-j)!Γ(-j+n-q-m+α+32), then, with the aid of Zeilberger’s algorithm [39], S_{n, q, m} satisfies the following first-order recurrence relation:

(18) (q+m)(2n-2q-2m+2α+1)Sn,q,m-(m+1)(2n-2m+2α-1)Sn,q,m+1=0,

with the initial value Sn,q,0=Γ(n+α+12)Γ(n-q+α+32).

The exact solution for this recurrence relation is Sn,q,m=(q+m-1)!Γ(n-m+α+12)(q-1)!m!Γ(n-q-m+α+32).

This proves Lemma 3.3.

Now, we are in a position to state and prove the desired derivatives formula.

Theorem 3.1. The derivatives of the normalized shifted Jacobi polynomials J̃n(α,α+1)(x) of any degree and for any order in terms of J̃n(α,α+1)(x) are given explicitly by

(19) DqJ̃n(α,α+1)(x)=∑m=0[n-q2]Ãn,m,qJ̃n-2m-q(α,α+1)(x)+∑m=0[n-q-12]B̃n,m,qJ̃n-2m-q-1(α,α+1)(x),

where

(20) Ãn,m,q=22qn!(q+m-1)!Γ(n-m+α+32)Γ(n-q-2m+2α+2)(b-a)qm!(q-1)!(n-q-2m)!Γ(n-q-m+α+32)Γ(n+2α+2),

and

(21) B̃n,m,q=22qn!(q+m)!Γ(n-m+α+12)Γ(n-q-2m+2α+1)(b-a)qm!(q-1)!(n-q-2m-1)!Γ(n-q-m+α+32)Γ(n+2α+2).

Proof. We prove the theorem for [a,b]≡[-1,1], that is, we prove the following formula:

(22) DqJn(α,α+1)(x)=∑m=0[n-q2]An,m,qJn-2m-q(α,α+1)(x)+∑m=0[n-q-12]Bn,m,qJn-2m-q-1(α,α+1)(x),

where

(23) An,m,q=2qn!(q+m-1)!Γ(n-m+α+32)Γ(n-q-2m+2α+2)m!(q-1)!(n-q-2m)!Γ(n-q-m+α+32)Γ(n+2α+2),

and

(24) Bn,m,q=2qn!(q+m)!Γ(n-m+α+12)Γ(n-q-2m+2α+1)m!(q-1)!(n-q-2m-1)!Γ(n-q-m+α+32)Γ(n+2α+2).

If we differentiate Eq. (10) q-times with respect to x, and noting the identity Dqxi=(i-q+1)qxi-q, then, we get

(25) DqJn(α,α+1)(x)=∑k=0[n-q2]en,k,qxn-2k-q+∑k=0[n-q-12]fn,k,qxn-2k-q-1,

where

(26) en,k,q=(-1)kn!Γ(α+1)2n-2k+2α+1Γ(n-k+α+32)πk!Γ(n+2α+2)(n-2k-q)!,

(27) fn,k,q=(-1)k+1n!Γ(α+1)2n-2k+2αΓ(n-k+α+12)πk!Γ(n+2α+2)(n-2k-q-1)!.

Making use of relation (15), enables one to write DqJn(α,α+1)(x)=∑1+∑2, where ∑1=∑k=0[n-q2]en,k,q∑i=0[n-q2]-kci,n-2k-qJn-q-2k-2i(α,α+1)(x)+∑k=0[n-q-12]fn,k,q∑i=0[n-q2]-k-1di,n-2k-q-1Jn-q-2k-2i-2(α,α+1)(x), and ∑2=∑k=0[n-q2]en,k,q∑i=0[n-q-12]-kdi,n-2k-qJn-q-2k-2i-1(α,α+1)(x)+∑k=0[n-q-12]fn,k,q∑i=0[n-q-12]-kci,n-2k-q-1Jn-q-2k-2i-1(α,α+1)(x).

Expanding ∑1 and ∑2 and collecting similar terms, then after some rather manipulation, we get DqJn(α,α+1)(x)=∑m=0[n-q2]An,m,qJn-2m-q(α,α+1)(x)+∑m=0[n-q-12]Bn,m,qJn-2m-q-1(α,α+1)(x), where

(28) An,m,q=∑j=0m{en,j,qcm-j,n-q-2j+fn,j,qdm-j-1,n-q-2j-1},

and

(29) Bn,m,q=∑j=0m{en,j,qdm-j,n-q-2j+fn,j,qcm-j,n-q-2j-1}.

Now, it is not difficult to show that en,j,qcm-j,n-q-2j+fn,j,qdm-j-1,n-q-2j-1=(-1)j2q-1n!(2n-2m+2α+1)Γ(n-j+α+12)Γ(n-q-2m+2α+2)j!(m-j)!(n-q-2m)!Γ(n+2α+2)Γ(n-j-q-m+α+32), and en,j,qdm-j,n-q-2j+fn,j,qcm-j,n-q-2j-1=(-1)j2qn!(q+m)Γ(n-j+α+12)Γ(n-q-2m+2α+1)j!(m-j)!Γ(n+2α+2)(n-q-2m-1)!Γ(n-j-q-m+α+32).

Finally, making use of Lemma 3.3, A_{n, m, q} and B_{n, m, q} take the forms An,m,q=2qn!(q+m-1)!Γ(n-m+α+32)Γ(n-q-2m+2α+2)m!(q-1)!(n-q-2m)!Γ(n-q-m+α+32)Γ(n+2α+2), and Bn,m,q=2qn!(q+m)!Γ(n-m+α+12)Γ(n-q-2m+2α+1)m!(q-1)!(n-q-2m-1)!Γ(n-q-m+α+32)Γ(n+2α+2).

Replacing x in (22) by (2x-a-bb-a), formula (19) can be obtained. This completes the proof of Theorem 3.1.

Remark 3.1. It is to be noted here that relation (19) may be written in the alternative form

(30) DqJ̃n(α,α+1)(x)=∑i=0n-qẼn,i,qJ̃i(α,α+1)(x),

where

(31) Ẽn,i,q=22qn!Γ(i+2α+2)(b-a)qi!(q-1)!Γ(n+2α+2){(n-i+q-22)!Γ(n+i+q+2α+32)(n-i-q2)!Γ(n+i-q+2α+32),(n+i+q)even,(n-i+q-12)!Γ(n+i+q+2α+22)(n-i-q-12)!Γ(n+i-q+2α+42),(n+i+q)odd.

Remark 3.2. As a direct consequence of relation (30), the qth derivative of J̃n(α+1,α)(x) can be easily deduced. This result is given in the following theorem.

Theorem 3.2. The derivatives of the normalized Jacobi polynomials J̃n(α+1,α)(x) of any degree and for any order in terms of J̃n(α+1,α)(x) are given explicitly by

(32) DqJ̃n(α+1,α)(x)=∑m=0[n-q2]F̃n,m,qJ̃n-2m-q(α+1,α)(x)+∑m=0[n-q-12]G̃n,m,qJ̃n-2m-q-1(α+1,α)(x),

where F̃n,m,q and G̃n,m,q are given as: F̃n,m,q=n-2m-q+α+1n+α+1Ãn,m,q,G̃n,m,q=-(n-2m-q+α)n+α+1B̃n,m,q, and Ãn,m,q and B̃n,m,q are given, respectively, in (20) and (21), or alternatively in the form