Two Parameter Gamma Function and its Properties

0.09 MB·

by  Kuldeep Singh Gehlot

Two Parameter Gamma Function and It’s Properties
Kuldeep Singh Gehlot
January 5, 2017
Government College Jodhpur,
JNV University Jodhpur, Rajasthan, India-306401.
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Email: drksgehlot@rediffmail.com
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Abstract

n
a
J In this paper we introduce the Two Parameter Gamma Function, Beta Function and Pochham-

3 mer Symbol. We named them, as p - k Gamma Function, p - k Beta Function and p - k

] Pochhammer Symbol and denoted as pΓk(x), pBk(x,y) and p(x)n,k respectively. We prove
A the several identities for Γ (x), B (x,y) and (x) those satisfied by the classical Gamma,
p k p k p n,k
C Beta and Pochhammer Symbol. Also we provide the integral representation for the Γ (x) and
p k
. B (x,y).
h p k
t
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m Mathematics Subject Classification : 33B15.
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Keywords: Two Parameter Pochhammer Symbol, Two Parameter Gamma Function, Two
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Parameter Beta Function, Two Parameter Psi Function, p - k Hypergeometric Function.
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0 1 Introduction
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The main aim of this paper is to introduce Two Parameter Pochhammer Symbol, Two Pa-
.
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rameter Gamma Function and Two Parameter Beta Function. p - k Gamma Function is the
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7 deformation of the classical Gamma Function, such that pΓk(x) ⇒ kΓk(x) = Γk(x) as p = k
1 and Γ (x) ⇒ Γ (x) = Γ(x) as p,k → 1.
p k 1 1
:
v In section 2, we defined Two Parameter Pochhammer Symbol denoted as p(x)n,k, with its con-
i vergent conditions. Two Parameter Pochhammer Symbol is the deformation of the classical
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Pochhammer Symbol, such that (x) ⇒ (x) = (x) as p = k and (x) ⇒ (x) =
r p n,k k n,k n,k p n,k 1 n,1
a (x) as p = k = 1. Also we derived two parameter Pochhammer symbol in terms of the ele-
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mentary symmetric function, and evaluate it’s derivative identities. It is most natural to relate
the two parameter Pochhammer symbol to the two parameter Gamma Function is defined. We
evaluate integral representation of two parameter Gamma Function, also represent two parame-
ter Gamma Function into different infinite product forms and so many recurrence relations are
evaluated.
In section 3, we defined two parameter Beta Function and two parameter Psi Function. Also
evaluate some recurrence relations and functional relation with classical Beta Function.
Section 4, deal with the definition of Hypergeometric function with Two Parameter Pochham-
merSymbol,knownasp-kHypergeometricfunction. AlsoweevaluatetheDifferentialEquation,
Functional relation with Classical Hypergeometric function and Integral Representation of p-k
Hypergeometric function.
Throughout this paper Let C,R+,Re(),Z−andN be the sets of complex numbers, positive real
numbers, real part of complex number, negative integer and natural numbers respectively.
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2 p - k Pochhammer Symbol and p - k Gamma Function
Inthis section we introducep- k Pochhammer Symboland p- k GammaFunction. We evaluate
Γ (x) in terms of limit, recurrence formulas and infinite products.
p k
2.1 Definition
Let x ∈ C;k,p ∈ R+ −{0} and Re(x) > 0,n ∈ N, the p - k Pochhammer Symbol (i.e. Two
Parameter Pochhammer Symbol), (x) is given by
p n,k
xp xp xp xp
(x) = ( )( +p)( +2p).........( +(n−1)p). (2.1)
p n,k
k k k k
For s,n ∈N with 0≤ s ≤ n, the sth elementary symmetric function
en(x ,x ,.....,x ) = x .....x
s 1 2 n i1 is
X
1≤i1≤i2≤...≤is≤n
on the variables x ,x ,.....,x .
1 2 n
Theorem 2.1 Formula for the p - k Pochhammer Symbol (i.e. Two Parameter Pochhammer
Symbol) in terms of the elementary symmetric function is given by
n−1
x
(x) = pnen−1(1,2,...,(n−1))( )n−s. (2.2)
p n,k s k
X
s=0
Where x ∈ C;k,p ∈ R+−{0} and Re(x) > 0,n ∈N.
Proof: The well known identity for elementary symmetric polynomials appear when expand a
linear factorization of a monic polynomial
n n−1
(λ+X ) = λnen−1+λn−1en−1+.....+λen−1 = en−1(1,2,...,(n−1))(λ)n−s. (2.3)
j 0 1 n−1 s
Y X
j=1 s=0
Using equation (2.1), we have the desired result.
Theorem 2.2 The derivative identities for p - k Pochhammer Symbol.
n−1
∂ n p
[ (x) ] = − (x) + s (x) (x+(s+1)k) . (2.4)
p n,k p n,k p n,k p n−1−s,k
∂k k k
X
s=1
∂ n
[ (x) ] = (x) . (2.5)
p n,k p n,k
∂p p
Where x ∈ C;k,p ∈ R+−{0} and Re(x) > 0,n ∈N.
Proof: Using the definition (2.1) and logarithmic derivatives, we have the desired result.
2.2 Definition
For x ∈ C/kZ−;k,p ∈ R+ −{0} and Re(x) > 0,n ∈ N, the p - k Gamma Function (i.e. Two
Parameter Gamma Function), Γ (x) is given by
p k
1 n!pn+1(np)kx
Γ (x) = lim . (2.6)
p k
k n→∞ p(x)n+1,k
or
1 n!pn+1(np)kx−1
Γ (x) = lim . (2.7)
p k
k n→∞ p(x)n,k
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Theorem 2.3 Given x ∈ C/kZ−;k,p,s,r ∈ R+ − {0} and Re(x) > 0,n ∈ N, the following
identities holds,
kx
(x) = ( ) . (2.8)
p n,s p n,k
s
p kx
(x) = ( )n ( ) . (2.9)
p n,s s n,k
s s
p
(x) = ( )n (x) . (2.10)
p n,k s n,k
s
k kx
Γ (x) = Γ ( ). (2.11)
p s p k
s s
k r kx
x
rΓs(x) = ( )s pΓk( ). (2.12)
s p s
r
x
rΓk(x) = ( )k pΓk(x). (2.13)
p
Proof: Property (2.8), (2.9), (2.10) follows directly from definition (2.1) and the results (2.11),
(2.12), (2.13) will follow directly by using equation (2.6).
Theorem 2.4 Given x ∈ C/kZ−;k,p ∈ R+ − {0} and Re(x) > 0, then the integral repre-
sentation of p - k Gamma Function is given by
pΓk(x) = ∞e−tpktx−1dt. (2.14)
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Proof: Consider the right hand side integral and ([3], Page 2) Tannery’s Theorem and equation
(2.7), we have
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∞e−tpktx−1dt = lim (np)k(1− tk )ntx−1dt.
Z n→∞Z np
0 0
Let A (x),i = 0,1,...,n, be given by
n,i
1
(np)k tk
A (x)= (1− )i tx−1dt.
n,i
Z np
0
Integration by parts we have the following recurrence formula,
ki
A (x) = A (x+k).
n,i n,i−1
pxn
Also,
A (x) = (np)k1 tx−1dt = (np)xk .
n,0 Z x
0
Therefor,
1 n!pn+1(np)xk−1
A (x)= .
n,n k (x) (1+ x )
p n,k kn
1 n!pn+1(np)kx−1
Γ (x)= lim A (x) = lim .
p k n,n
n→∞ k n→∞ p(x)n,k
Which complete the proof.
Theorem 2.5 Given x ∈ C/kZ−;k,p ∈ R+−{0} and Re(x) > 0, then we have,
x ∞
pΓk(x) = pk [(1+ 1)xk(1+ x )−1]. (2.15)
k n nk
Y
n=1
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Proof: Using equation (2.1) and (2.7), we immediately get the desire result.
Theorem 2.6 Given x ∈ C/kZ−;k,p ∈ R+−{0} and Re(x) > 0, then we have,
n
1 x x
pΓk(x) = kpxk nl→im∞[n−kx Y(1+ rk)]. (2.16)
r=1
Proof: Using equation (2.1) and (2.7), we immediately get the desire result.
Theorem 2.7 Given x ∈ C/kZ−;k,p ∈ R+−{0} and Re(x) > 0, then we have,
pΓk(x) = akx ∞e−tpkatx−1dt. (2.17)
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Proof: Using equation (2.14), we immediately get the desire result.
Theorem 2.8 Given x ∈ C/kZ−;k,p ∈ R+−{0} and Re(x) > 0, then we have,
∞
1 x x
pΓk(x) = kpxk exkγ Y[(1+ nk)e−nxk]. (2.18)
n=1
Where
1 1
γ = lim [1+ +....+ −logn],
n→∞ 2 n
is Euler’s constant.
Proof: Using equation (2.16), we immediately get the desire result.
Theorem 2.9 The relation between p - k Gamma Function, k-Gamma Function and classi-
cal Gamma Function is given by,
x
p x pk x
pΓk(x) = ( )kΓk(x) = Γ( ). (2.19)
k k k
Where x ∈ C/kZ−;k,p ∈ R+−{0} and Re(x) > 0,
Proof: Using (2.14) and Proposition 4, page 3 of [1], we get the desire result.
Theorem 2.10 For x ∈ C/kZ−;n,q ∈ N;k,p ∈ R+ − {0} and Re(x) > 0, then the rela-
tion between p - k Pochhammer Symbol, k-Pochhammer Symbol and classical Pochhammer
Symbol is given by,
p x
(x) = ( )n(x) = (p)n( ) . (2.20)
p n,k n,k n
k k
Also for Generalized p - k Pochhammer Symbol, we have
p x q x +r−1
(x) = ( )nq (x) = (p)nq ( ) = (pq)nq (k ) . (2.21)
p nq,k nq,k nq n
k k q
Y
r=1
Proof: Using (2.1), (2.19) and Proposition 4, page 3 of [1], we get the desire result.
Theorem 2.11 For x ∈ C/kZ−;k,p ∈ R+ − {0} and Re(x) > 0,n ∈ N. The fundamental
equations satisfied by p - k Gamma Function, Γ (x) are,
p k
Γ (x+nk)
p k
(x) = . (2.22)
p n,k
Γ (x)
p k
xp
Γ (x+k) = Γ (x). (2.23)
p k p k
k
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x x x
Γ (x+nk)= pn( )( +1)......( +(n−1)) Γ (x). (2.24)
p k p k
k k k
Γ (x) pn
p k
= (x−k)(x−2k).....(x−nk). (2.25)
Γ (x−nk) kn
p k
Γ (x) Γ (−x+nk+k)
p k = (−1)n p k . (2.26)
Γ (x−nk) Γ (−x+k)
p k p k
1
pk 1
Γ (1) = Γ( ). (2.27)
p k
k k
p
Γ (k) = . (2.28)
p k
k
p
pk p
Γ (p)= Γ( ). (2.29)
p k
k k
π 1
Γ (x) Γ (−x) = . (2.30)
p k p k xk sin(πx)
k
p π
Γ (x) Γ (k−x) = . (2.31)
p k p k k2 sin(πx)
k
m−1 m−1
pΓk(x+ kmr)= kpm2−1(2π)(m2−1)m12−mkx pΓk(mx);m = 2,3,4,.... (2.32)
Y
r=0
Proof: All the results follow directly from using equation (2.1), (2.7) and (2.14).
Theorem 2.12 For x ∈ C/kZ−;k,p ∈ R+ −{0} and Re(x) > 0,n ∈ N. Then the recurrence
relations for p - k Pochhammer Symbol are given by,
n (x) = (x) − (x−k) . (2.33)
p n−1,k p n,k p n,k
And
(x) = (x) × (x+jk) . (2.34)
p n+j,k p j,k p n,k
Proof: Usingequation(2.20)andbasicrelationsn(x) = (x) −(x−1) ,(x) = (x) (x+j) ,
n−1 n n n+j j n
we get the desired result.
3 p - k Beta Function and p - k Psi Function
In this section, we introduce the p - k Beta Function B (x,y) and p - k Psi Function ψ (x,y).
p k p k
We evaluate explicit formula that relate the B (x,y) and ψ (x) to classical Beta function
p k p k
B(x,y) and Classical Psi function respectively ψ(x). Also prove some identities.
3.1 Definition
The p - k Beta Function B (x,y) is given by
p k
Γ (x) Γ (y)
p k p k
B (x,y) = ;Re(x) > 0,Re(y) >0. (3.1)
p k
Γ (x+y)
p k
Theorem 3.1 The B (x,y) function satisfies the following identities.
p k
pBk(x,y) = k1 Z 1tkx−1(1−t)yk−1dt. (3.2)
0
1 1 tkx−1+tky−1
B (x,y) = dt. (3.3)
p k k Z x+y
0 (t+1) k
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∞
pBk(x,y) = Z tx−1(1+tk)−x+kydt. (3.4)
0
1 x y
B (x,y) = B( , ). (3.5)
p k
k k k
Proof: Using the definition (3.1), we have immediately above results.
3.2 Definition
The logarithmic derivative of the p - k Gamma Function is known as p - k Psi Function, ψ (x).
p k
d 1 d
ψ (x) = ln[ Γ (x)] = [ Γ (x)]. (3.6)
p k p k p k
dx Γ (x)dx
p k
x
ln[ Γ (x)] = ψ (x)dx. (3.7)
p k Z p k
1
Theorem 3.2 Some properties of ψ (x) are given by
p k
lnp x
ψ (x) = +ψ( ). (3.8)
p k
k k
∞
lnp k 1
ψ (x) = −γ− +x . (3.9)
p k
k x n(x+nk)
X
n=1
∞
lnp 1
ψ (x) = −γ+(x−k) . (3.10)
p k
k (n+1)(x+nk)
X
n=0
Where γ is Euler’s Constant and ψ(x) is Classical Psi Function.
Proof: Using the definition (3.2), we have immediately above results.
Theorem 3.3 The rth derivative of p - k Psi Function, ψ (x) yields the result in terms of
p k
k-Zeta Function, ζ (x,r),
k
dr dr−1 ∞ 1
[ln[ Γ (x)]] = ψ (x) = (−1)rk(r−1)! ,for r ≥ 2 (3.11)
dxr p k dxr−1 p k (x+nk)r
X
n=0
Where k-Zeta Function given by definition 15, page 8 of [1].
∞
1
ζ (x,r) = .
k (x+nk)r
X
n=0
Proof: Using the definition (3.2) and differentiate, we get the desired result.
4 Hypergeometric Function
In this section we define the Hypergeometric Function using p - k Pochhammer Symbols. Here
we are use the notation of [2].
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4.1 Definition
Given x ∈ C, a ∈ Cr;k,p ∈ (R+)r;s,t ∈ (R+)q,b = (b ,b ,...,b ) ∈ Cq such that b ∈ C/s Z−.
1 2 q i i
The p-k hypergeometric function F(a,p,k;b,t,s;x) is given by
∞ r (a ) xn
F(a,p,k;b,t,s;x) = i=1 pi i n,ki . (4.1)
Qq (b ) n!
nX=0 j=1 tj j n,sj
Q
By using Ratio Test we can show that the series (4.1) converges for all finite x if r ≤ q. If
r > q+1, the series diverges and if r = q+1, it converges for all x such that |x| < | t1t2.....tq |.
p1p2.....pr
Theorem 4.1 Given x ∈ C, a ∈ Cr;k,p ∈ (R+)r;s,t ∈ (R+)q,b = (b ,b ,...,b ) ∈ Cq such
1 2 q
that b ∈ C/s Z−. Then the Functional relation between p - k Hypergeometric Function and
i i
Classical Hypergeometric Function is given by,
r
a b p
F(a,p,k;b,t,s;x) = F( ; ; i=1 ix). (4.2)
k s Qq t
j=1 j
Q
Proof: Using definition (2.20), we get above result.
Theorem 4.2 The Differential Equation of p - k Hypergeometric Function is given by
q r
b a
j i
[θ (θ+ −1)−Ax (θ+ )]W = 0. (4.3)
s k
Y j Y i
j=1 i=1
r
p
Where θ = x d , A = i=1 i and W = F(a,p,k;b,t,s;x).
dx Qq t
j=1 j
For r ≤ q +1, i = 1,2Q,...,r and j = 1,2,...,q when no bj is a negative integer or zero and no
sj
two bj is differ by an integer or zero.
sj
Proof: Using Function relation (4.2), we get the desired result.
Theorem 4.3 For any a ∈C;k,p > 0 and |x| < 1, the following identity holds
p
∞ (a) xn
p n,k = (1−xp)−ak. (4.4)
n!
X
n=0
Proof: Using (2.20), we get immediately the desired result.
Theorem 4.4 Given x ∈ C, a ∈ Cr;k,p ∈ (R+)r;s,t ∈ (R+)q,b = (b ,b ,...,b ) ∈ Cq such
1 2 q
that b ∈ C/s Z−. The Integral Representation of p - k Hypergeometric Function is given by,
i i
F(a,p,k;b,t,s;x) = r q Γ(sbjj) 1tkaii−1(1−t)sbjj−kaii−1eptjixtdt. (4.5)
Yi=1jY=1Γ(akii)Γ(bsjj − akii)Z0
Proof:Using (4.2), we get immediately the desired result.
References
[1] Diaz, R. and Pariguan, E. On hypergeometric functions and Pochhammer k-symbol. Divul-
gaciones Mathematicas, Vol. 15 No. 2 (2007) 179-192.
[2] Earl D. Rainville, Special Function, The Macmillan Company, New york,1963.
[3]Erdelyi,A., HigherTranscendentalFunction Vol. 1,McGraw-Hill Book Company,NewYork,
1953.
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