In 2007, Díaz and Pariguan introduced the one-parameter deformation of the classical gamma function in the form [7]:

Γp(z)=∫0∞ t z−1e− tp p dt,

where z∈ℂ,ℜ(z)>0 and p>0. In fact, the occurrence of the product of the form x(x+p)(x+2p)⋯(x+(n−1)p) in the combinatorics of creation and annihilation operators [6, 8] and the perturbative computation of Feynman integrals [5] led them to generalize the gamma function in the above stated form and to generalize the Pochhammer p-symbol in the following form:

(z)n,p=z(z+p)(z+2p)⋯(z+(n−1)p),

in which z∈ℂ, p∈ℝ and n∈ℕ. These generalizations lead to the following elementary properties.

Γp(z+p)=zΓp(z),Γp(p)=1,(z)k,p=Γp(z+kp)Γp(z),(z)n−k,p=(−1)k(z)n,p(p−z−np)k,p,(z)mn,p=mmn∏j=1mz+jp−pmn,p.

For p>0, a∈ℂ and |x|<1p, Diaz et al. [7] demonstrated that

(1.1)∑ n=0∞ (a) n,pn!xn=(1−px)−ap.

This may be regarded as the p-deformed binomial series. The radius of convergence of this series can be enlarged or diminished by choosing a smaller or larger p; unlike in the classical theory of the radius of convergence of the binomial series which is fixed. This motivated us to study the p-deformation of certain Special functions, particularly, the polynomial system formed by generalized Humbert polynomials.

Our objective is to extend generalized Humbert polynomials according to Gould [9] by involving a new parameter: p(>0). We call this extension a p-deformation of the polynomial. We study its properties, namely, the differential equation, generating function relations (GFRs), differential recurrence relations, and mixed relations, and we illustrate the companion matrix.

The companion matrix of the monic polynomial is defined as follows.

Definition 1.1.

Let f(x)∈ℂ[X] be a monic polynomial given by f(x)=δ0+δ1x+δ2x2+⋯+δk−1xk−1+xk. Then the k×k matrix, called the companion matrix of f(x), is denoted and defined as follows [14, p. 39]:

C(f(x))=010⋯0001⋯0⋮⋮⋮⋮000⋯1−δ0−δ1−δ2⋯−δk−1.

We have the following lemma [14, Proposition 1.5.14, p.39].

Lemma 1.1.

If f ∈ K[x] is nonconstant and A=C(f(x)), then f(A)=O, the null matrix.

The class {Pn(m,x,γ,s,c);n=0,1,2,…} of generalized Humbert polynomials is defined below [9.Eq.5.11, p.707]

Definition 1.2.

For m∈ℕ,n∈ℕ∪{0}, and x∈ℝ,

(1.2)Pn(m,x,γ,s,c)=∑ k=0 [n/m]s−n+mkksn−mkcs−n+mk−k×γk (−mx)n−mk, 

where γ, c, and s are generally unrestricted.

This class of polynomials is generated by the following relation:

(1.3)(c−mxt+γtm)s=∑ n=0∞Pn(m,x,γ,s,c)tn.

The substitution s=-ν, γ=1, and c=1 in (1.2) result in Humbert polynomials, according to Humbert [11]:

(1.4)Πn,mν(x)=∑ k=0 n/m(−mx)n−mkΓ(1−ν−n+(m−1)k)(n−mk)! k!.

Apart from the polynomials (1.4), according to Humbert [12, p. 75], Humbert functions also exits. They have explicit representations in a double infinite series, given by the following:

Ψ1(a;b;c,d;x,y)=∑r=0∞∑s=0∞ (a) r+s   (b)r (c)r   (d)s  r! s!xrys,Ψ2(a;b,c;x,y)=∑r=0∞∑s=0∞ (a) r+s (b)r   (c)s  r! s!xrys,Ξ1(a,b;c;d;x,y)=∑r=0∞∑s=0∞ (a)r   (b)s   (c)r (d) r+s  r! s!xry

and

Ξ2(a,b;c;x,y)=∑ r=0∞∑s=0∞ (a)r  (b)r (c) r+s r! s!xrys,

where |x|<1, |y|<∞ and c, d≠0,−1,−2,…. In recent years, these functions have been increasingly used in various fields, for example, in theoretical physics [3, 13] and communication theory [1, 2, 18]. Moreover, for specific values of parameters and variables, their reduced forms have also been found useful, especially in connection with simplification algorithms in computer algebra systems (see [3]). Therefore, we propose the following extension of the polynomials Pn(m,x,γ,s,c).

Definition 1.3.

For γ, s, c∈ℂ, m∈ℕ, x∈ℝ, n∈ℕ∪{0}, and p>0,

(1.5)Pn,p(m,x,γ,s,c)=∑ k=0 n/mγkcs−n+mk−k(s+p)mk−k−n,p(n−mk)! k! (−mx)n−mk,

in which the floor function r=floor r, represents the greatest integer ≤ r.

We call these polynomials p-deformed generalized Humbert polynomials or pGHPs. When p=1, it coincides with the polynomial (1.2). The particular polynomials belonging to these general p-polynomials provide an extension to the Humbert polynomials (1.4), Kinney polynomials, Pincherle polynomials, Gegenbauer polynomials, and Legendre polynomials (see [9]). For instance, the substitutions γ=1, c=1, and s=-ν in (1.5) yield p-deformed Humbert polynomials:

(1.6)Πn,m,pν(x)=∑ k=0 n/m(−mx)n−mkΓp(p−ν−np+(m−1)kp)(n−mk)! k!.

For p=1, this coincides with (1.4). If we substitute ν=1/m, m∈ℕ, in (1.6), then we obtain the p-deformed Kinney polynomial:

Pn,p(m,x)=∑ k=0 n/m(−mx)n−mkΓp(p−1/m−np+(m−1)kp)(n−mk)! k!.

For m=3 and ν=1/2, (1.6) reduces to the p-deformed Pincherle polynomial:

Pn,p(x)=∑ k=0 n/3(−3x)n−3kΓp(p−1/2−np+2kp)(n−3k)! k!.

The p-deformed Gegenbauer polynomial is the special cases where m=2 of (1.6) which occurs in the following form:

(1.7)Cn,pν(x)=∑ k=0 n/2(−2x)n−2kΓp(p−ν−np+kp)(n−2k)! k!.

Further, if ν=1/2 then (1.7) is reduced to the p-deformed Legendre polynomial given by the following:

(1.8)Pn,p(x)=∑ k=0 n/2(−2x)n−2kΓp(p−1/2−np+kp)(n−2k)! k!.

All these polynomials reduce to their classical forms when p=1 [9, p.697].

In this section, we derive the differential equation of the polynomial (1.5). Costa et al. [4] demonstrated that the homogeneous differential equation:

(1−xN)y(N)+∑ k=1 NAN−kxN−ky(N−k)=0,

has a polynomial solution if and only if 0≤r<N, ∀r, and ∃ n≥0 such that n mod N =r, where n is a root of the recurrence relation, and y(j) is the jth derivative of y with respect to x for j=1,2,…,N.

Let the sequence {fr}r=0n be given by f_r=f(r), where

f(r)=(n−r)−s+rp+n−rmpm−1,p.

We use the forward difference operator 'Δ' and the shift operator 'E' which are defined as follows [10, Eq. (5.2.13), p. 178]:

Δft=ft+1−ft, Ekft=ft+k.

The relation between Δ and E is given by [10, Eq. (5.2.14), p. 178] Δ=E−1, where 1 is the identity operator defined by 1f=f. In (1.5), we use the following formula:

(p+s)−n+mk−k,p=(p+s)−(n−mk+k),p=(−1)n−mk+k(p−p−s)n−mk+k,p,

to obtain the alternative form:

(2.1)Pn,p(m,x,γ,s,c)=∑ k=0 n/m(−1)kγkcs−n+mk−k(−s)n−mk+k,p(n−mk)! k!(mx)n−mk.

The differential equation for this explicit form is derived in Theorem 2.1.

Theorem 2.1.Let s∈ℂ, p>0, and m∈ℕ. Then, the polynomial y=Pn,p(m,x,γ,s,c) satisfies the following equation:

(2.2)γcm−1y(m)+∑ r=0marxry(r)=0,

where ar=mm−1r!Δrf0.

Proof. Let n=ml+q, where n/m=l and 0≤q≤m−1. The r^th derivative of (2.1) is given by the following:

DrPn,p(m,x,γ,s,c)=∑ k=0 (n−r)/m(−1)kγkcs−n+mk−k(−s)n−mk+k,p(n−mk−r)! k!mn−mk×xn−mk−r. 

Hence,

(2.3)xrDrPn,p(m,x,γ,s,c)=∑ k=0 (n−r)/m(−1)kγkcs−n+mk−k(−s)n−mk+k,p(n−mk−r)! k!×(mx)n−mk. 

Next, we have

(2.4)DmPn,p(m,x,γ,s,c)=∑ k=0 (n−m)/m(−1)kγkcs−n+mk−k(−s)n−mk+k,p(n−mk−m)! k!×mn−mk xn−mk−m =∑ k=0 l−1(−1)kγkcs−n+mk−k(−s)n−mk+k,p(n−mk−m)! k!mm ×(mx)n−mk−m, 

where

n−rm=l,if r≤ql−1,if r>q.

Substituting the equation (2.3) and (2.4) on the left-hand side of the differential equation (2.2) and comparing the corresponding coefficients of x, we find that

(2.5)∑ r=0mn−mkrr!ar = m mk(−s)n−mk+k+m−1,p(−s)n−mk+k,p  = mmk(−s+np−mkp+kp)m−1,p, 

where k=0,1,2,…,l−1, and

∑ r=0qn−mlrr!ar =mml(−s+np−mlp+lp)m−1,p. 

Because n=ml+q⇒n−ml=q, we have the following:

(2.6)∑ r=0qqrr!ar =mm−1(n−q)−s+qp+ n−qmpm−1,p.

By substituting ar=mm−1Δrf0r! in (2.6), we obtain

∑ r=0qqrΔrf0 =(n−q)−s+qp+ n−qmpm−1,p,

that is,

(2.7)(1+Δ)qf0 = (n−q)−s+qp+n−qmpm−1,p.

Hence, 1+Δ=E, the shift operator, in (2.7) becomes the following:

Eqf0 = f(q) = (n−q)−s+qp+n−qmpm−1,p.

For k=0,1,2,…,l−1, (2.5) can be written in the following form:

(2.8)∑ r=0mn−mkrΔrf0 = fn−mk = mk(−s+np−mkp+kp)m−1,p.

As t↦f(t) is a polynomial of degree m, the equation (2.8) is a forward difference formula for f at the point t=n-mk. Thus, the proof is completed for the choice:

ar=mm−1Δrf0r!=mm−1r!Δrnnp−msm m−1,p.

2.1. Particular Cases

We illustrate special instances of the differential equation (2.2). In particular, the equations of the Pincherle, Gegenbauer, and Legendre polynomials. We choose r=0, 1, 2 and 3 to obtain the following:

(2.9)a0=mm−1Δ0f0 = mm−1nnp−msmm−1,p, (2.10)a1=mm−1Δf0 = mm−1(E−1)f(0) = mm−1(f(1)−f(0))=mm−1(n−1)(n−1)p+m(−s+p)mm−1,p−nnp−msmm−1,p, (2.11)a2=mm−1Δ2f02! = mm−12!(E−1)2f0 = mm−12!(E2−2E+1)f(0)=mm−12!(f(2)−2f(1)+f(0))=mm−12!(n−2)(n−2)p+m(−s+2p)mm−1,p−2(n−1)×(n−1)p+m(−s+p)mm−1,p+nnp−msmm−1,p,

and

(2.12)a3=mm−1Δ3f03! = mm−13!f(3)−3f(2)+3f(1)−f(0)=mm−13!(n−3)(n−3)p+m(−s+3p)m m−1,p−3(n−2)× (n−2)p+m(−s+2p) m m−1,p+3(n−1)×(n−1)p+m(−s+p)m m−1,p−nnp−msm m−1,p.

By choosing m=3 in (2.9), (2.10), (2.11) and (2.12), we obtain the following:

(2.13)a0=32nnp−3s32,p = n(np−3s)(np−3(s−p)), (2.14)a1=32(n−1)(n−1)p+3(−s+p)32,p−(n)np−3s32,p=3np(np−2s+p)−(3s−2p)(3s−5p), (2.15)a2=12ps−18p2, (2.16)a3=−4p2.

Further, after entering m=3, γ=1, c=1, and s=-λ into the differential equation (2.2), from the particular values (2.13), (2.14), (2.15) and (2.16), we arrive at the differential equation of the p-deformed Pincherle polynomial. From the following general form:

y(3)+∑ r=03arxry(r)=0,

that is,

(1+a3x3)y(3)+a0y+a1xy(1)+a2x2y(2) = 0,

we obtain the following equation:

1−4p2x3y(3)−62pλ+3p2x2y(2)+3np(np+2λ+p)−(3λ+2p)(3λ+5p)xy(1)+n(np+3λ)(np+3(λ+p))y = 0.

The choice p=1 yields the differential equation of the Pincherle polynomial according to Humbert [11,p. 23].

Next, to obtain the equation for the p-deformed Gegenbauer polynomial, we enter m=2, γ=1, c=1, and s=-ν into (2.2) to obtain

y(2)+∑ r=02arxry(r)=0,

or equivalently,

(1+a2x2)y″+a1xy′+a0y = 0.

From (2.9), (2.10) and (2.11), we have the following:

a0=nnp−2s,a1=2(n−1)(n−1)p+2(−s+p)21,p−(n)np−2s21,p=2(n−1)(n−1)p+2(−s+p)2−(n)np−2s2 = 2s−p,a2=−p.

With a₀, a₁, and a₂, the above equation takes the precise form:

(1−px2)y″+n(np+2ν)y−(2ν+p)xy′ = 0,

where y=Cn,pν(x) is given by (1.7). When p=1, this reduces to the differential equation of the Gegenbauer polynomial [17, Eq.(1.4), p.279]. The well-known special case ν=1/2 of this equation is the following differential equation:

(1−px2)Pn,p′′(x)−(1+p)xPn,p′(x)+n(np+1)Pn,p(x)=0

of the p-deformed Legendre polynomial (1.8). In addition, for p=1, this reduces to the differential equation of the Legendre polynomial P_n(x) (cf. [17, Eq.(5), p.161]).

We derive GFR of the pGHP (1.5). This is accomplished with the help of the p-deformed version of the identity [16, Ex. 212 and 216, p. 146]:

(1+z)a+11−zb=∑ n=0∞a+bn+nnwn,

where a,b∈ℂ and w=z(1+z)−b−1. It is given as explained in Theorem 3.1.

Theorem 3.1. For p>0, a,b∈ℂ, and w=z(1+z)b+1.

(3.1)(1+z)a/p+11−zb=∑ n=0∞Γp(a+bnp+np+p)Γp(a+bnp+p) n! wnpn.

Proof. We use the Lagrange's series [19, Eq. (3), p. 354]:

f(z)1−wg′(z)=∑ n=0∞wnn!Dznf(z)(g(z))nz=z0,  D=ddz,

where w=z−z0g(z).

To derive (3.1), we take z0=0,f(z)=(1+z)a/p,g(z)=(1+z)b+1, and w=z(1+z)−b−1 in the left-hand side of Lagrange's series gives the following:

f(z)1−wg′(z)=(1+z)a/p1−w(b+1)(1+z)b=(1+z)a/p1−z(1+z)(b+1)=(1+z)a/p+11−zb.

The same substitution on the right-hand side of the Lagrange's series gives the following:

∑n=0∞wnn!Dznf(z)(g(z))nz=z0=∑n=0∞wnn!Dzn(1+z)ap+bn+nz=0=∑n=0∞wn n!ap+bn+nap+bn+n−1×⋯ap+bn+n−n+1=∑n=0∞wn(−1)npn n!(−a−bnp−np)n,p=∑n=0∞wnpn n!(a+bnp+p)n,p=∑n=0∞Γp(a+bnp+np+p)Γp(a+bnp+p) n!wnpn.

This completes the proof.

We define the function as follows:

(3.2)R(An,α,γ,r,m,p) = ∑ k=0 n/mΓp (−α+mrk+p)Γp (−α+mrk−kp+p) k!γkp−kAn−mk,

which is required in deriving the following GFR.

For m∈ℕ, w=t(1+γwm)−β/p, p>0, G(z)=∑ n=0∞Anzn where A0≠0,

(3.3)∑ n=0∞R(An,α+βn,γ,r,m,p)tn=(1+γwm)(p−α)/p1+ βmp+1γwmGw (1+γwm ) r/p,

where {A_n} is an arbitrary sequence such that ∑ i=0∞|Ai|<∞ and other parameters are generally unrestricted.

Proof. We begin with the following:

∑n=0∞R(An,α+βn,γ,r,m,p)tn=∑n=0∞∑k=0n/m Γp (p−α−βn−nr+mrk) Γp (p−α−βn−nr+mrk−kp) k!γkp−kAn−mktn=∑n=0∞∑k=0∞ Γp (p−α−βn−βmk−nr) Γp (p−α−βn−βmk−nr−kp) k!γkp−kAntn+mk=∑n=0∞Antn∑k=0∞ Γp(p−α−βn−βmk−nr) Γp(p−α−βn−βmk−nr−kp) k!γkp−ktmk.

In view of the sum in (3.1), the inner series simplifies to the following:

∑n=0∞R(An,α+βn,γ,r,m,p)tn=∑n=0∞An tn (1+v) (−α−βn−nr)/p+1 1+ βmp+1v= (1+v) (p−α)/p 1+ βmp+1v∑n=0∞An t (1+v) −β/p n (1+v) nr/p,

where v=γtm(1+v)−βm/p. If we replace v with γ w^m, then w=t(1+γwm)−β/p and, consequently, we obtain

∑n=0∞R(An,α+βn,γ,r,m,p)tn= (1+γwm) (p−α)/p1+ βmp+1γwm∑n=0∞An t (1+γwm ) −β/p n (1+γwm) nr/p= (1+γwm) (p−α)/p1+ βmp+1γwm∑n=0∞An w (1+γ w m )r/p n.

This completes the proof of GFR (3.3).

The replacement of γ with γ p/c, and the substitutions α=−s,r=p, and

An = (−m)n cs−nΓp(p+s)Γp(p+s−np) n! xn

in (3.3), yield the GFR of the pGHP in the following form:

(3.4)∑ n=0∞Pn,p(m,x,γ,s+βn,c)tn=(1+γpwm/c)(p+s)/p1+γpwmc βmp+1Gw 1+ γpwm c ,

where w=t1+γpwmc−β/p and

G(u) = ∑ n=0∞(c)s−nΓp(p+s)Γp(p+s−np)n!(−mxu)n.

We note that β=0⇔w=t; and hence, using the p-binomial series (1.1), we find the following:

(3.5)∑n=0∞Pn,p(m,x,γ,s,c)tn=(1+γptm/c)s/p Gt1+γptmc=c(1−1/p)s (c+γptm)s/p∑n=0∞(−m)nΓp(p+s)Γp(p+s−np)n!×xtc+γptmn=c(1−1/p)s(c+γptm)s/p∑n=0∞(−m)n(p+s)−n,pn!×xtc+γptmn=c(1−1/p)s(c+γptm)s/p∑n=0∞(−s)n,pn!×mxtc+γptmn=c(1−1/p)s(c+γptm)s/p1−pmxtc+γptms/p=c(1−1/p)sc+γptm−pmxts/p.

This generalizes the GFR (1.3).

The GFR of the p-deformed Humbert polynomials occurs as a special case of (3.4) with the substitutions γ=1, c=1, and s=−μ given by

∑ n=0∞Πn,m,pμ+βn(x)tn=(1+pwm)(p−μ)/p1+βm+pwmGw 1+pwm ,

where w=t1+pwm−β/p and

G(z) = ∑ n=0∞Γp(p−μ)Γp(p−μ−np)n!(−mxz)n.

The case β =0 yields the following GFR:

(3.6)∑ n=0∞Πn,m,pμ(x)tn=1+ptm −pmxt−μ/p.

This extends the GFR according to Humbert [11, p.24] (also see [15, Eq.(1.15), p.5] with p=1).

The GFR of the p-deformed Kinney polynomial occurs with γ=1, c=1, and s=-1/m from (3.4) given by

∑ n=0∞Pn,p(m,βn,x)tn=(1+pwm)(p−1/m)/p1+βm+pwmGw 1+pwm ,

where w=t1+pwm−β/p and

G(z)=∑ n=0∞Γp(p−1m)Γp(p−1m−np)n!(−mxz)n.

The GFR of the p-deformed Pincherle polynomial is obtained by substituting m=3, γ=1, c=1, and s=-λ in (3.4), and it is given by the following:

∑ n=0∞Pn,pλ+βn(x)tn=(1+pw3)(p−λ)/p1+ β3p+1pw3Gw 1+pw3 ,

where w=t1+pw3−β/p and

G(z) = ∑ n=0∞Γp(p−λ)Γp(p−λ−np)n!(−3xz)n.

Similarly, taking m=2, γ=1, c=1, and s= -ν in (3.4), we obtain

(3.7)∑ n=0∞Cn,pν+βn(x)tn=(1+pw2)(p−ν)/p1+ β2p+1pw2Gw 1+pw2 ,

where w=t1+pw2−β/p and

G(z) = ∑ n=0∞Γp(p−ν)Γp(p−ν−np) n!(−2xz)n,

which is the GFR of the p-deformed Gegenbauer polynomial. Furthermore, entering ν=1/2 into (3.7), we obtain the GFR of the p-deformed Legendre polynomial or briefly pLP given by

∑ n=0∞Pn,p(x)tnn=(1+pw2)(2p−1)/2p1+ β2p+1pw2Gw 1+pw2 ,

with w=t1+pw2−β/p and

G(z) = ∑ n=0∞Γp(p−12)Γp(p−12−np) n!(−2xz)n.

3.1. Fibonacci-type Polynomials

We provide a computation formula of Fibonacci-type polynomials of order n (cf. [15, Theorem 2.2, p.6]) in Theorem using (3.5).

Theorem 3.3. For the pGHP defined by (1.5),

(3.8)Pn,p(m,x,γ,s1+s2,c)=∑ k=0nPn−k,p(m,x,γ,s1,c) Pk,p(m,x,γ,s2,c).

Proof. Replacing s with s₁+s₂ in GFR (3.5) provides the following result:

∑n=0∞Pn,p(m,x,γ,s1+s2,c)tn=(c)(1−1/p)(s1+s2)c+γptm−pmxts1+s2/p=(c)(1−1/p)s1c+γptm−pmxts1/p×(c)(1−1/p)s2c+γptm−pmxts2/p=∑n=0∞Pn,p(m,x,γ,s1,c)tn∑k=0∞Pk,p(m,x,γ,s2,c)tk=∑n=0∞∑k=0∞Pn,p(m,x,γ,s1,c)Pk,p(m,x,γ,s2,c)tn+k=∑n=0∞∑k=0nPn−k,p(m,x,γ,s1,c)Pk,p(m,x,γ,s2,c)tn.

Comparing the coefficients of tⁿ, this yields (3.8).

The GFR (3.6) leads to the computation formula of Fibonacci-type polynomials of order n, stated as Corollary 3.4.

Corollary 3.4. In the usual notations and meaning,

Πn,m,pμ1+μ2(x)=∑ k=0nΠn−k,m,pμ1(x)Πk,m,pμ2(x).