DOI 10.1007/s00601-012-0461-8
M. K. Bahar · F. Yasuk
Approximate Solutions to the Dirac Equation with Effective
Mass for the Manning–Rosen Potential in N Dimensions
Received: 11 April 2012 / Accepted: 10 May 2012 / Published online: 8 June 2012 © Springer-Verlag 2012
Abstract The solutions of the effective mass Dirac equation for the Manning–Rosen potential with the centrifugal term are studied approximately in N dimension. The relativistic energy spectrum and two-com-ponent spinor eigenfunctions are obtained by the asymptotic iteration method. We have also investigated eigenvalues of the effective mass Dirac–Manning–Rosen problem for α = 0 or α = 1. In this case, the Manning–Rosen potential reduces to the Hulthen potential.
1 Introduction
Quantum mechanical systems with spatially dependent mass have been received great interests in recent years [1–29]. These systems proved to be useful in modelling the physical and electronic properties of semiconduc-tors [1], quantum wells and quantum dots [2,3], quantum liquids [4] and semiconductors heterostructures [5]. The analytical solutions of the non-relativistic Schrödinger equation with position-dependent mass for solv-able potentials have been addressed by a number of methods [6–17]. Moreover, the investigation of relativistic effects is important in quantum mechanical systems such as heavy atoms and heavy ion doping [19–21]. Thus, the investigation of the relativistic Dirac equation with position dependent mass is very important for these systems. In the recent literature, many studies have been devoted to investigate of the exact or quasi-exact solutions of the Dirac equation with position dependent mass and properties of these solutions for different potentials and mass distributions [18–29]. The solutions of the N-dimensional non-relativistic Schrödinger and relativistic Klein–Gordon and Dirac equations with exactly and quasi-exactly solvable potentials have been investigated over the past decades [30–41].
In this paper, we have investigated the approximate solutions of the N-dimensional Dirac equation with spa-tially dependent mass in the presence of the Manning–Rosen potential proposed by Manning and Rosen [42,43]. This exponential-type potential is important diatomic molecular potential model. The energy eigenvalues and corresponding eigenfunctions have been obtained by using exponential approximation of the centrifugal term within the framework of the asymptotic iteration method (AIM) which is based on solving second-order homogeneous linear differential equations [44–46].
The organization of this paper is as follows: in the Sect.2, the asymptotic iteration method is given shortly. For the position dependent mass Dirac equation with the Manning–Rosen potential, the relativistic energy eigenvalues and normalized eigenfunctions are presented and some numerical results and grahps are given in M. K. Bahar· F. Yasuk (
B
)Department of Physics, Erciyes University, 38039 Kayseri, Turkey E-mail: yasuk@erciyes.edu.tr
M. K. Bahar
Department of Physics, Karamanoglu Mehmetbey University, 70100 Karaman, Turkey
the obtained results are compared with the ones before. Finally, concluding remarks and discussions are given in the last section.
2 The Asymptotic Iteration Method
We briefly outline the AIM here; the details can be found in Refs. [44–46]. The AIM was proposed to solve second-order differential equations of the form
y= λ0(x)y+ s0(x)y (1)
whereλ0(x) = 0 and s0(x), λ0(x) are in C∞(a, b). The variables, s0(x) and λ0(x), sufficiently differentiable. The differential equation (1) has a general solution
y(x) = exp ⎛ ⎝− x αdx ⎞ ⎠ ⎡ ⎢ ⎣C2+ C1 x exp ⎛ ⎜ ⎝ x λ0(x) + 2α(x) d x ⎞ ⎟ ⎠dx ⎤ ⎥ ⎦ (2)
if, for sufficiently large n,
sn λn = sn−1 λn−1 = α (3) where λn(x) = λn−1(x) + sn−1(x) + λ0(x)λn−1(x) sn(x) = sn−1(x) + s0(x)λn−1(x), n = 1, 2, 3 . . . (4) The termination condition of the method together with Eq. (4) can be also written as follows:
δ(x) = λn−1(x)sn(x) − λn(x)sn−1(x) = 0 (5) For a given potential, the idea is to convert the relativistic wave equation to the form of Eq. (1). Then, s0 andλ0are determined and sn andλn parameters are calculated. The energy eigenvalues are obtained by the termination condition given by Eq. (5). However, the exact eigenfunctions can be derived from the following wave function generator:
yn(x) = C2exp ⎛ ⎝− x αkd x ⎞ ⎠ (6)
where n= 0, 1, 2, . . . and k is the iteration step number and usually greater than n.
3 Approximate Solutions of the N-Dimensional Dirac–Manning–Rosen Problem with Position-Dependent Mass
The N-dimensional Dirac equation with a central potential V(r) and position-dependent mass μ(r) can be written in natural units ¯h = c = 1 as [39,47]
Hψ(r) = Enrκψ(r) where H = N j=1 αjpj+ βμ(r) + V (r) (7) where Enrκis the relativistic energy,αj and β are Dirac matrices, which satisfy anticommutation relations
αjαk+ αkαj = 2δj k αjβ + β αj = 0 (8) αj2= β2= 1
and
pj = −i∂j = −i ∂
∂xj
1≤ j ≤ N (9)
The orbital angular momentum operators Lj k, the spinor operators Sj k and the total angular momentum operators Jj kcan be defined as follows:
Lj k = −Lj k = ixj ∂ ∂xk − ix k ∂ ∂xj, S j k = −Sk j = i αjαk/2, Jj k = Lj k+ Sj k. L2= N j<k L2j k, S2= N j<k S2j k, J2= N j<k Jj k2, 1 ≤ j < k ≤ N. (10)
For a spherically symmetric potential, total angular momentum operator Jj k and the spin-orbit operator
K = −β(J2− L2− S2+ (N − 1)/2) commutate with the Dirac Hamiltonian. For a given total angu-lar momentum j , the eigenvalues of K areκ = ±( j + (N − 2)/2) for unaligned spin j = + 12 and κ = ( j + (N − 2)/2) for unaligned spin j = −1
2. Also, since V(r) is spherically symmetric, the symmetry group of the system is SO(N) group.
Thus, we can introduce the hyperspherical coordinates x1= r cos θ1
xα = r sin θ1· · · sin θα−1cosφ, 2≤ α ≤ N − 1
xN = r sin θ1· · · sin θN−2sinφ where the volume element of the configuration space is given as
N j=1 d xj = rN−1dr d d = N−1 j=1 (sin θj)j−1dθj (11) with 0≤ r < ∞, 0 ≤ θk ≤ π, k = 1, 2, . . . N − 2, 0 ≤ φ ≤ 2π, such that the spinor wavefunctions can be classified according to the hyperradial quantum number nr and the spin-orbit quantum numberκ and can be written using the Pauli–Dirac representation
ψnrκ(r, N) = f g = r−N−1 2 Fnrκ(r)Y j m( N) i Gnrκ(r)Yj m˜ ( N) (12)
where Fnrκ(r) and Gnrκ(r) are the radial wave function of the upper and the lower-spinor components respec-tively, Y j m( N) and Yj m˜ ( N) are the hyperspherical harmonic functions coupled with the total angular momentum j . The orbital and the pseudo-orbital angular momentum quantum numbers for spin symmetry and pseudospin symmetry ˜ refer to the upper and lower-component respectively [29].
Inserting Eq. (12) into the first of Eq. (7), and separating the variables we obtain the following coupled radial Dirac equation for the spinor component:
d dr + κ r Fnrκ(r) = [μ(r) + Enrκ− V (r)]Gnrκ(r) (13) d dr − κ r Gnrκ(r) = [μ(r) − Enrκ+ V (r)]Fnrκ(r) (14)
whereκ = ±(2 + N − 1)/2. Using Eq. (13) as the upper component and substituting into Eq. (14), we obtain the following second-order differential equations
dr2 − r2 − (μ(r) + Enrκ− V (r))(μ(r) − Enrκ+ V (r)) − dμ(r) dr − d V(r) dr d dr +κr (μ(r) + Enrκ− V (r)) ⎤ ⎦ Fnrκ(r) = 0 (15) d2 dr2 − κ(κ − 1) r2 − (μ(r) + Enrκ− V (r))(μ(r) − Enrκ+ V (r)) − dμ(r) dr + d V(r) dr d dr −κr (μ(r) − Enrκ+ V (r)) ⎤ ⎦ Gnrκ(r) = 0 (16) The energy eigenvalues in these equations depend on the angular momentum quantum number and dimension N. To solve these equations, we can use an approximation for the centrifugal barrier as discussed in the following.
In the following sub-sections, with in the framework of the AIM, we construct the relativistic energy spectrum and corresponding eigenfunctions of the position-dependent mass Dirac equation with the Manning–Rosen potential in N-dimension by using exponential approximation of the centrifugal term.
3.1 Energy eigenvalues of the Dirac equation with Position-Dependent Mass for the Manning–Rosen potential
In this study, to obtain the analytical approximate solutions of the Dirac equation for a single particle with position dependent mass in the Manning–Rosen potential with spin-orbit coupling term, we have to use an approximation in the following form [34,37,38]:
1 r2 ∼
e−r/b
b2(1 − e−r/b)2 (17)
which is only valid for large values of the parameter b. We consider the Manning–Rosen potential defined as,
V(r)= 1 b2 α(α −1)e−2r/b (1 − e−r/b)2 − Ae−r/b 1− e−r/b , = 2μ/¯h2 (18)
where A and α are two dimensionless parameters [48,49], but parameter b has dimension of length and is screening parameter for the potential. It is shown that this potential remains invariant by mappingα ↔ 1 − α and has a minimum value V(r0) = − A
2
4b2α(α−1)at r0= bI n
1+2α(α−1)A forα > 1.
Equation (15) can not be solved analytically because of the last term in the equation, so, we use the equality dμ(r)
dr − dV(r)
dr = 0 to eliminate this term. Thus, using this equality condition, the mass function is obtained as the following: μ (r) = μ0+ 1 b2 α(α −1)e−2r/b (1 − e−r/b)2 − Ae−r/b 1− e−r/b (19) whereμ0is the integral constant and corresponds to the rest mass of the Dirac particle. The mass functionμ(r) has the same form as the Manning–Rosen potential. As can be seen above, Eq. (19) becomes singular near the origin. The reason of this case is structure of the Manning–Rosen potential. For these reasons, method of investigation of this problem can be better understood by examination of Fig.1. The Manning–Rosen potential and mass function is inserted Eq. (15), we get
d2 dr2 − κ(κ + 1) r2 − (μ0+ Enrκ) μ0− Enrκ+ 2 b2 α(α − 1)e−2r/b (1 − e−r/b)2 − Ae−r/b 1− e−r/b Fnrκ(r) = 0 (20)
Fig. 1 The variation of the Manning–Rosen potential according to various values ofα screening parameter and r. (1/b =
0.025, A = 2b, = 2, α = 0.01 black line, α = 0.2 red line, α = 0.5 blue line, α = 0.9 green line) (colour figure online)
The r12 ∼
e−r/b
b2(1−e−r/b)2 exponential approximation is used for the centrifugal term and defined z = e−r/b transformation, Eq. (20) becomes
d2 dz2 + 1 z d dz − ξ0 z(1 − z)2− ξ1 z2 + ξ2 (1 − z)2+ ξ3 z(1 − z) Fnrκ(z) = 0 (21) where areξ0= κ(κ +1), ξ1= b2(μ02− E2nrκ), ξ2= 2(μ0+Enr κ) α(α −1) andξ3= 2(μ0+E nr κ)A.
In order to solve Eq. (21) with the aid of the AIM, we should transform Eq. (21) to the form of Eq. (1). The wave function should respect the boundary conditions, i.e. Fnrκ(0) = 0 at z=0 for r → ∞ and Fnrκ(1) = 0 at z=1 for r→ 0. Therefore, the reasonable physical wave function is proposed as follows:
Fnrκ(z) = z √ξ
1(1 − z)12(1+γ)fn
rκ(z) (22)
where isγ = √1+ 4ξ0+ 4ξ2. If Eq. (22) is inserted into Eq. (21), the second-order homogeneous linear differential equations in the following form are obtained:
fnrκ(z) = −−1 − 2 √ ξ1+ z(2 + 2√ξ1+ γ) z(−1 + z) f nrκ(z) − 1+ 2ξ0+ γ +2√ξ1(1 + γ) − 2ξ3 2z(−1 + z) fnrκ(z) (23) whereλ0(z) = −−1+z+2(−1+z) √ξ1+z(1+γ ) z(−1+z) , s0(z) = 1+2ξ0+γ +2√ξ1(1+γ )−2ξ3 2z(1−z) .
By means of Eq. (4), we may calculateλn(z) and sn(z). This gives:
λ1(z) = (4 + 8(−1 + z) 2ξ 1+ 6(−1 + z)√ξ1(−2 + z(3 + γ )) 2(−1 + z)2z2 − z(8 − 2ξ0+ 3(1 + γ ) + 2ξ3) +z2(4 − 2ξ 0+ 5(1 + γ ) + 2(1 + γ )2+ 2ξ3)), (24) s1(z) = (−2 + 2(−1 + z) √ ξ1+ z(4 + γ ))(1 + 2ξ0+ γ + 2√ξ1(1 + γ ) − 2ξ3) 2z2(−1 + z)2 (25) . . .etc.
Combining these results obtained by the AIM with quantization condition given by Eq. (5) yields: s0λ1− s1λ0= 0 ⇒ ξ10= (1 + 2ξ0+ γ − 2ξ3) 2 4(1 + γ )2 , (26a) s1λ2− s2λ1= 0 ⇒ ξ11= (2 + 2ξ0+ 3(1 + γ ) − 2ξ3) 2 4(3 + γ )2 , (26b) s2λ3− s3λ2= 0 ⇒ ξ12= (8 + 2ξ0+ 5(1 + γ ) − 2ξ3) 2 4(5 + γ )2 . . . (26c)
the case of position dependent mass
α = 0.25 α = 0.75 α = 1
1/b= 0.025 1/b= 0.050 1/b= 0.075
|n, l, κ Energy |n, l, κ Energy |n, l, κ Energy
N=3 |0, 0, 1 0.586071 |0, 0, 1 0.605847 |0, 0, 1 0.657298 |1, 0, 1 0.807668 |1, 0, 1 0.828670 |1, 0, 1 0.861394 |2, 1, 2 0.943282 |2, 1, 2 0.961914 |2, 1, 2 0.978113 |3, 2, 3 0.980288 |3, 2, 3 0.993706 |3, 2, 3 0.999733 |4, 3, 4 0.993923 |4, 3, 4 – |4, 3, 4 – N=5 |0, 0, 2 0.814032 |0, 0, 2 0.835110 |0, 0, 2 0.861394 |1, 0, 2 0.901102 |1, 0, 2 0.921319 |1, 0, 2 0.942049 |2, 1, 3 0.966780 |2, 1, 3 0.983064 |2, 1, 3 0.994225 |3, 2, 4 0.988713 |3, 2, 4 0.998698 |3, 2, 4 – |4, 3, 5 0.997157 |4, 3, 5 – |4, 3, 5 – N=7 |0, 0, 3 0.902231 |0, 0, 3 0.922516 |0, 0, 3 0.942049 |1, 0, 3 0.943931 |1, 0, 3 0.962531 |1, 0, 3 0.978113 |2, 1, 4 0.980413 |2, 1, 4 0.993794 |2, 1, 4 0.999733 |3, 2, 5 0.993954 |3, 2, 5 – |3, 2, 5 – |4, 3, 6 0.999003 |4, 3, 6 – |4, 3, 6 –
If the Eq. (26) are generalized, the indirect energy eigenvalue of the position dependent mass Dirac equation for the Manning–Rosen potential in N-dimension is as follows:
ξ1n= 1 4 2n2+ (2n + 1)(1 + γ ) + 2ξ0− 2ξ3 2n+ 1 + γ 2 (27) Ifξ0andξ3statements are inserted into Eq. (27) and is compared withξ1, the energy eigenvalues can be obtained by the following expression:
En2= μ20− 1 4b2 ⎡ ⎢ ⎢ ⎣ 2n2+ (2n + 1) 1+ 1+ 4k(k + 1) +8(m0+E n)α(α−1) + 2k(k + 1) − 4(m0+En)A 1+ 2n + 1+ 4k(k + 1) + 8(m0+E n)α(α−1) ⎤ ⎥ ⎥ ⎦ 2 (28) When the dimension of the system is N = 3, N = 5 and N = 7 for different α, b values and quantum states, the energy eigenvalues are calculated from Eq. (28). The results are shown in Table1.
As shown Table1, due to increase values ofα screening parameter, Eneigenvalues also increase. Because, as shown in Fig. 1, the Manning–Rosen potential well become deep for increasing values of α screening parameter. In this case, the more bound states occur in the potential well.
3.2 Eigenfunctions of the Dirac equation with Position-Dependent Mass for the Manning–Rosen potential Also, the corresponding eigenfunctions can be obtained for the position dependent mass Dirac equation with Manning–Rosen potential by using the wave function generator given in Eq. (6). The first lowest states are:
f0(z) = 1 f1(z) = −C2(1 + 2 ! ξ1) 1−2+ 2 √ ξ1+ γ 1+ 2√ξ1 z , f2(z) = C2(1 + 2 ! ξ1)(2 + 2 ! ξ1) 1−2(3 + 2γ √ ξ1) (1 + 2√ξ1) z+2(3 + 2γ √ ξ1)(4 + 2γ√ξ1) (1 + 2√ξ1)(2 + 2√ξ1) z2 , f3(z) = −C2(1 + 2 ! ξ1)(2 + 2 ! ξ1)(3 + 2 ! ξ1) 1−6(2 + γ √ ξ1) (1 + 2√ξ1) z+(2 + γ √ ξ1)(5 + 2γ√ξ1) (1 + 2√ξ1)(1 +√ξ1) z2
−(4 + 2γ √ ξ1)(5 + 2γ√ξ1)(3 + γ√ξ1) (1 + 2√ξ1)(1 +√ξ1)(3 + 2√ξ1) z3 (29) . . .etc.
Thus, the wave function fnκ(z) can be written as:
fnκ(z) = (−1)nC2(1 + 2 √ ξ1+ n) (1 + 2√ξ1) 2 F1(−n, 1 + 2 ! ξ1+ γ + n; 1 + 2 ! ξ1; z) (30) Hence, we can write the total radial wavefunction as follows:
Fnrκ(z) = Nz √ξ
1(1 − z)21(1+γ )2F1(−n, 1 + 2!ξ1+ γ + n; 1 + 2!ξ1; z) (31) In order to normalize the wavefunctions we should determine Gnrκ(z). Therefore, if we insert Eq. (17), (18) andμ (r) = μ0− 1 b2 α(α −1)e−2r/b (1−e−r/b)2 − Ae −r/b 1−e−r/b
into Eq. (16), we obtain that d2 dr2− κ(κ − 1) r2 − μ2 0− En2rκ− 2(μ0− Enrκ) b2 α(α − 1)e−2r/b (1 − e−r/b)2 − Ae−r/b 1− e−r/b Gnrκ(r)=0 (32) When 1/r2exponential approximation and defined z = e−r/btransformation are used, Eq. (32) becomes as follows: d2 d z2+ 1 z d d z − A ξ 2 z(1 − z)2 − ξ1 z2 + ξ 3 (1 − z)2 − ξ 4 z(1 − z) Gnrκ(z) = 0 (33) where are ξ2 = κ(κ − 1), ξ3 = 2(μ0−Enr κ)α(α−1)
, ξ4 =
2(μ0−Enr κ)A
. If the processes to find Fnrκ(z) are repeated, we obtain
Gnrκ(z) = Nz √ξ
1(1 − z)21(1+χ)2F1(−n, 1 + 2!ξ1+ χ + n; 1 + 2!ξ1; z) (34) where isχ = 1+ 4ξ2− 4ξ3.
The total radial wave functions are obtained in terms of confluent hypergeometric functions. In this part of our work, we calculate the normalization constant Nin Eqs. (31) and (34). Calculation of the normalization constant is very necessary, which has not been explicitly worked out in most of the studies. How to compute the normalization constant is given in some studies [35]. To compute the normalization constant N, firstly, we start with the normalization condition."0∞( f2+ g2)rN−1dr = 1 [47]. According to the transformation of z = e−r/b, z → 1 for r → 0 and z → 0 for r → ∞. Therefore,"01bF(z)z 2d z+"01bGz(z)2d z= 1. So, we have by means of the above normalization condition that is
(N)2 "1 0 z2 √ ξ1−1(1 − z)1+γ 2F1(−n, 1 + 2√ξ1+ γ + n; 1 + 2√ξ1; z) 2 d z +"1 0 z2 √ ξ1−1(1 − z)1+χ2F1(−n, 1 + 2√ξ1+ χ + n; 1 + 2√ξ1; z) 2d z = 1/b (35)
The series representation of the confluent hypergeometric function2F1is [50,51]
pFq(a1, . . . , ap; c1, . . . , cq; s) = ∞ i=0 (a1)i. . . (ap)i (c1)i. . . (cq)i si i!. (36)
Being Pochammer symbols(a1)i, (c1)iand a polynomial of degree n in s, we obtain the normalization constant in the following form:
(N)2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ n # i=0 (−n)i(1+2√ξ1+γ +n)i(2√ξ1)i (1+2√ξ1)ii!(2+2√ξ1+√1+4ξ0)iB(2 √ ξ1, 2 + γ ) 3F2(−n, 1 + γ + 2√ξ1+ n, 2√ξ1+ i; 1 + 2√ξ1, 2 + γ + 2√ξ1+ i; 1) +#n i=0 (−n)i(1+2√ξ1+χ+n)i(2√ξ1)i (1+2√ξ1)ii!(2+χ+2√ξ1)i B(2 √ ξ1, 2 + χ) 3F2(−n, 1 + χ + 2√ξ1+ n, 2√ξ1+ i; 1 + 2√ξ1, 2 + χ + 2√ξ1+ i; 1) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = 1/b (37)
energy eigenvalues of the Dirac–Hulthen problem with position dependent mass for N = 3, N = 5 and N = 7 1/b = α = 0.025, = 2, A = 2b, μ0= ¯h = 1
N= 3 Energy N= 5 Energy N= 7 Energy
|n, l, κ Manning Rosen Potential with effective mass Hulthen Pot. with effective mass in Ref. [52] |n, l, κ Manning Rosen Potential with effective mass Hulthen Pot. with effective mass in Ref. [52] |n, l, κ Manning Rosen Potential with effective mass Hulthen Pot. with effective mass in Ref. [52] |0, 0, 1 0.61969 0.61969 |0, 0, 2 0.82180 0.82180 |0, 0, 3 0.90464 0.90464 |1, 0, 1 0.82180 0.82180 |1, 0, 2 0.90464 0.90464 |1, 0, 3 0.94518 0.94518 |2, 1, 2 0.94518 0.94518 |2, 1, 3 0.96748 0.96748 |2, 1, 4 0.98071 0.98071 |3, 2, 3 0.98071 0.98071 |3, 2, 4 0.98889 0.98889 |3, 2, 5 0.99403 0.99403 |4, 3, 4 0.99403 0.99403 |4, 3, 5 0.99720 0.99720 |4, 3, 6 0.99902 0.99902 4 Discussions
It is known that the Manning–Rosen potential reduces to the Hulthen potential for α = 0 or α = 1. The Hulthen potential defined as;
V(r)= −Z α e − α r
1− e− α r (42)
whereα is the screening parameter and Z is the Coulomb charge. Now, we will check our results in the case ofα = 0 or α = 1, i.e, in the relativistic Hulthen limit. So, if taking bα = 1 and identifying A/ b as Z, we obtain as follows results:
As can be seen in Table2, forα = 0 or α = 1, there is an excellent agreement with the results. 5 Conclusions
To summarize, we have considered approximately analytical bound state solutions of the Dirac equation with the Manning–Rosen potential and mass function using the AIM by applying an approximation to the centrif-ugal like term, which is a different approach. For arbitrary spin-orbit quantum number,κ state and the small α, we have obtained the energy eigenvalues and corresponding radial wavefunctions for the case of position dependent mass and approximation on the spin-orbit coupling term in a systematic way. We must point out that numerical calculations for eigenenergies of the Dirac states in Eq. (28) are sensitive to the choice of the param-etersα and b. The corresponding eigenfunctions have been obtained in terms of confluent hypergeometric functions and they have also been normalized. Finally, we have investigated eigenvalues of position dependent mass Dirac–Manning–Rosen problem forα = 0 or α = 1. In this case, the Manning–Rosen potential reduces to the Hulthen potential. The results are related to Ref.[52] in the Table2.
Acknowledgments The authors are grateful to the anonymous referees for their illuminating criticism and suggestions.
Appendix
To normalize the wavefunctions, some of the special procedures for the beta function is given in the following form [50,51]: (i) Bq(x, y) = "q 0 z x−1(1 − z)y−1d z= 1, B(x, y) = (x)(y)/ (x + y), (ii) Bq(x + 1, y) = x+yx Bq(x, y) −q x(1−q)y x+y (iii) Iq(x, y) = Iq(x − 1, y) − qx−y(1 − q)y x+ y = Iq(x − 2, y) − qx−2(1 − q)y (x − 2)B(x − 2, y)− qx−1(1 − q)y (x − 1)B(x − 1, y)
= Iq(x − 3, y) − qx−3(1 − q)y (x − 3)B(x − 3, y)− qx−2(1 − q)y (x − 2)B(x − 2, y)− qx−1(1 − q)y (x − 1)B(x − 1, y) = . . . = Iq(x − m, y) − qx(1 − q)y m k=1 q−k (x − k)B(x − k, y) m= 1, 2, . . . (iv) Bq(x, y) = qx(1 − q)y−1 x ∞ k=0 (1 − y)k (1 + x)k q q− 1 k = qx(1 − q)y−1 x 2F1 1, 1 − y; 1 + x; q q− 1 for q∈ (−∞, 0) ∪ 0,1 2 and Bq(x, y) = B(x, y) − qx−1(1 − q)y y ∞ k=0 (1 − x)k (1 + y)k q− 1 q k = B(x, y)−qx−1(1−q)y y 2F1 1, 1−x; 1+y;q−1 q and B1(x, y)= B(x, y) for q=1 (v) (a)i+ j = (a)i(a + i)j References
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