Journal de Ciencia e Ingenierı́a, Vol.5, No.1, Agosto de 2013, pp. 37-41
Investigación - Matemáticas
Another Extension of the Little q-Jacobi Polynomial and its
Properties
Otra Extensión del q-Polinomio de Jacobi y sus Propiedades
Rajesh V. Savalia a ? B. I. Dave b J. C. Prajapati a ??
a Department
of Mathematical Sciences, Faculty of Applied Sciences, Charotar University of Science and Technololgy, Changa, Anand - 388
421, India
b Department of Mathematics, Faculty of Science, The Maharaja Sayajirao University of Baroda, Vadodara-390002, India.
Recibido: 5/2/2013; revisado: 16/4/2013; aceptado: 8/7/2013.
Savalia, R. V., Dave, B. I., Prajapati, J. C.: Another Extension of the Little q-Jacobi Polynomial and its
Properties. Jou.Cie.Ing. 5 (1): 37-41, 2013. ISSN 2145-2628.
Abstract
The little q-Jacobi polynomial is extended in this work whose properties such as fractional q-integral, qintegral representations, q-derivative, higher order q-derivative, q-Liebnitz’s formula, summation formula
are obtained. Also, the Fatou’s lemma of measurable function is applied on this polynomial.
Keywords: q-polynomial, q-fractional integral, q-derivative, q-fractional derivative, Lebesgue measure.
Resumen
En este trabajo se generaliza el q-polinomio de Jacobi y se obtienen sus propiedades q-integral fraccional,
q-representaciones integrales, q-derivada, q-derivadas de alto orden, q-fórmula de Leibnitz y la fórmula de
suma. Además, el lema de Fatou de función medible es aplicado a este polinomio.
Palabras Claves: q-polinomio, q-integral fraccional, q-derivada, q-derivada fraccional, medida de lebesgue.
1. Introducción
The little q-Jacobi polynomial is given by ([2], p.161, Eq.(7.1.11))


q −n , abq n+1 ; q, xq
,
pn (x ; a, b ; q) = 2 ϕ 1 
aq;
we consider an extension of this polynomial in the form
n /s
X
( q −n ; q)sk ( abq n+1 ; q)sk k k
r
pn (x ; s, a, b ; q) =
x q
( aq ; q)rk ( q ; q)k
k=0
? [email protected], [email protected]
[email protected]
(1)
(2)
Jou.Cie.Ing, Vol.5, No.1, Agosto de 2013, pp. 37-41
where |q| < 1, and n and s are positive integers, r is any integer, and a and b are complex numbers. For this
polynomial, we have obtained fractional q-integral formula, q-integral representations, q-derivative expression, higher
order q-derivative, q-Liebnitz’s formula, a summation formula and a result based on Lebesgue measure over R.
The following formulas and definitions are used in this paper [2].
The q-Factorial function is defined by,
(a ; q)n =





1 , if n = 0
(1 − a)(1 − aq 2 ) ... (1 − aq n−1 ) , if n = 1, 2, 3, ...




(a ;q)∞
(aq n ;q)∞ ,
(3)
if n ∈ C
The q-Beta function is given by
Z1
(1.4)
Bq (x, y) =
0
tx−1 (tq; q)∞
Γq (x)Γq (y)
dq t =
,
y
(tq ; q)∞
Γq (x + y)
(4)
The Kober fractional integral formula ([6], p.187, Eq.(1.1)), which is a generalization of the Riemann-Liouville
fractional integral formula ([6], p.187, Eq.(1.1)), is defined by
Iqη, µ f (x)
x− η − µ
=
Γq (µ)
Zx
(x − t q)µ − 1 tη f (t) dq t,
(5)
0
n
where (x − y)n = x (y/x ; q)n , Re( µ ) > 0, η ∈ R and |q| < 1.
Andrews and Askey’s integral is given by ([1], p.98, Eq.(1.12))
Z∞
0
tα−1
Γq (α) Γq (β) (−cq α , −q 1−α /c; q)∞
(−ctq α+β ; q)∞
dq t =
, Re(α, β) > 0
(−ct; q)∞
Γq (α + β) (−c, −q/c; q)∞
(6)
W. Hahn’s q-integral formula ([3], p.10, Eq,(3.16)) is defined by
Z∞
tλ−1 eq (−t) dq t =
0
(1 − q) (q; q)∞ −λ(λ−1)/2
q
,
(q λ ; q)∞
(7)
where Re(λ) > 0.
Here, we find the q-derivatives of the polynomial and write the expression for a general mth order. Now,
compared with a general formula
Dqm fn (x) =
1
m
xm (q − 1) q m(m−1)/2
m
X

j
(−1) q j(j−1)/2 
j=0
m
j

 fn (xq m−j )
(8)
q
this is due to D. S. Moak ([4], p.33, Eq.(6.9)), we obtained a particular summation formula connecting degree
extended little q-Jacobi polynomial 2 with the sum of its lower degrees. The formula 8 is also known as q-Leibnitz’s
formula. The following are the results obtained in this paper. On applying the Kober fractional integral formula 5
([6], p.187, Eq.(1.1)) to the q-polynomial (1.2), we have
µ
I η,
{pnr (x ; s, a, b ; q) } =
q
x
−1
n /s
Γq (µ − 1) X ( q −n ; q)sk ( abq n+1 ; q)sk (ηq ; q)k k k
q x
Γq (µ)
( aq ; q)rk ( q ; q)k (ηµ ; q)k
k=0
Next, the following q-integral representations are derived by making use of 6 and 7.
38
Rajesh V. Savalia et.al.:Another Extension of the Little q-Jacobi Polynomial and its Properties
prn (x ; s, a, b ; q) =
(1 − q)(aq : q)∞ (−c : q)∞ (−q/c : q)∞
(q : q)∞ (−abcq n+1 : q)∞ (−abq n /c : q)∞ (a2 b2 q n+1 : q)∞
Z∞
Bnr (x, ts ; s, a, b, c; q)
t( a + b + n)
dq t
(−ct ; q)∞ (−a2 b2 ctq n+1 ; q)∞
(9)
0
where
n /s
X
( q −n ; q)sk ( a2 b2 q n+1 ; q)sk ( −abcq n+1 ; q)sk k k sk
r
s
Bn (x, t ; s, a, b, c; q) =
x q t
( aq ; q)rk ( q ; q)k
k=0
prn (x
(abq n+1 ; q)∞
; s, a, b ; q) =
(1 − q)(q ; q)∞
where Jnr (x, ts ; s , a, b ; q) =
n /
Ps
k=0
Z∞
t( a + b + n) eq (− t) Jnr (x, ts ; s , a, b ; q) dq t ,
(10)
0
( q −n ; q)sk
xk q k tsk
k ( q ; q)k
q (a+b+n+sk+1)(a+b+n+sk)/2 ( aq ; q)
Following formula is derived using 8 as

m−1
Y

j=0
(q
−n+sj
n+sj+1
; q)s (abq
(aq jr+1 ; q)r
m

; q)s 
prn−ms (x ; aq mr , bq m(2s−r) ; q) =

m
X
m
(11)

(−1)
i
(−1) q i(i−1)/2   prn−ms (xq m−i ; s , a , b ; q)
xm q m(m−1)/2 i=0
i
q
where ms ≤ n.
Finally, by using the Fatou’s lemma [[5], p.86] we arrived at,
∞
“If
R {fn }1 isRa sequence of non negative measurable functions and fn → f almost everywhere on a set E , then
f ≤ lim fn ”.
E
n→∞ E
Following inequality
Z
Z
Vr (x ; s , a , b ; q) dm ≤ lim
n→ ∞
E
prn (xq sn ; s, a, b ; q) dm,
(12)
E
is derived where X = [ − ∞, 0], M = power set of X, m (E) = lenght of E, ∀E ∈ Ω, x ∈ X, a > 0, b > 0, r
is an integer, s and n are positive integers, 0 < q < 1, Vr (x ; s , a , b ; q) is an absolutely convergent q-series (as
a 0 Φrr - series) for every x which is given by
Vr (x ; s , a , b ; q) =
∞
k
X
(−1) q sk(sk−1)/2
k=0
(aq ; q)r k (q ; q)k
xk q k = lim prn (xq n ; s, a, b ; q)
n→∞
2. Proof of the Results
Using 2 and 5, we get
n /s
Zx
x− n − µ X ( q −n ; q)sk ( abq n+1 ; q)sk k
n, µ r
Iq pn (x ; s, a, b ; q) =
q
(x − tq)µ − 1 tn+k dq t
Γq (µ)
( aq ; q)rk ( q ; q)k
k=0
0
39
Jou.Cie.Ing, Vol.5, No.1, Agosto de 2013, pp. 37-41
n /s
Z1
x− η− µ X ( q −n ; q)sk ( abq n+1 ; q)sk k η+k+µ−1
(yq)µ − 1 y η+k dq y
=
q x
Γq (µ)
( aq ; q)rk ( q ; q)k
k=0
(13)
0
n /s
−1
x Γq (µ − 1) X ( q −n ; q)sk ( abq n+1 ; q)sk (ηq ; q)k k k
=
q x
Γq (µ)
( aq ; q)rk ( q ; q)k (ηµ ; q)k
k=0
Now, using 2 and 6, we get
prn (x ; s, a, b ; q) =
(1 − q)(aq : q)∞ (−c : q)∞ (−q/c : q)∞
(q : q)∞ (−abcq n+1 : q)∞ (−abq n /c : q)∞ (a2 b2 q n+1 : q)∞
Z∞
t( a + b + n)
0
Bnr (x, ts ; s, a, b, c; q)
dq t
(−ct ; q)∞ (−a2 b2 ctq n+1 ; q)∞
(14)
where
Bnr (x, ts ; s, a, b, c; q) =
n /s
X
( q −n ; q)sk ( a2 b2 q n+1 ; q)sk ( −abcq n+1 ; q)sk k k sk
x q t
( aq ; q)rk ( q ; q)k
k=0
The following q-integral representation is obtained
by making use of 2 and 7
prn (x
(abq n+1 ; q)∞
; s, a, b ; q) =
(1 − q)(q ; q)∞
Z∞
t( a + b + n) eq (− t) Jnr (x, ts ; s , a, b ; q) dq t
0
where
Jnr (x, ts ; s , a, b ; q) =
n /s
X
q (a+b+n+sk+1)(a+b+n+sk)/2
k=0
( q −n ; q)sk
xk q k tsk .
( aq ; q)k ( q ; q)k
Next, the q-derivative operator is defined [2] as
Dq (f (x)) =
f (xq) − f (x)
xq − x
The q-derivative of the polynomial 2 is
n /s
X
( q −n ; q)sk ( abq n+1 ; q)sk q k
r
Dq {pn (x ; s, a, b ; q)} =
Dq xk
( aq ; q)rk ( q ; q)k
k=0
−n+s
(q
; q)s (abq n+s+1 ; q)s
1
=
p n−s (x ; aq r , bq (2s−r) ; q)
(1 − q)
(aq; q)r
Similarly, second order q-derivative is given by


1
−n+sj
n+sj+1
Y
(q
; q)s (abq
; q)s 
1

Dq2 {prn (x ; s, a, b ; q)} =
p n−2s (x ; aq 2r , bq 2(2s−r) ; q)
2
jr+1 ; q)
(aq
(1 − q)
r
j=0
In general, if sm < −n , then mth order q-derivative is given by
40
(15)
Rajesh V. Savalia et.al.:Another Extension of the Little q-Jacobi Polynomial and its Properties
Dqm
{prn (x


m−1
Y (q −n+sj ; q) (abq n+sj+1 ; q)
1
s
s

; s, a, b ; q)} =
p n−ms (x ; aq mr , bq m(2s−r) ; q).
m
jr+1 ; q)
(aq
(1 − q)
r
j=0
(16)
On making use of 8 and 16, the summation formula obtained as


m−1
Y (q −n+sj ; q) (abq n+sj+1 ; q)
s
s r

pn−ms (x ; aq mr , bq m(2s−r) ; q) =
jr+1 ; q)
(aq
r
j=0
m

m
X
m

(−1)
i
(−1) q i(i−1)/2   prn−ms (xq m−i ; s , a , b ; q)
xm q m(m−1)/2 i=0
i
(17)
q
We know that a polynomial is a continuous function and hence, is measurable function [5]. Thus the polynomial
2 is a measurable function. If {fn (x)} is sequence of measurable functions and converges to f (x) then f (x) is
measurable function [5]. Now, we have
prn (xq n ; s, a, b ; q) → Vr (x ; s , a , b ; q)
∞
P
(−1)k q sk(sk−1)/2 k k
where Vr (x ; s , a , b ; q) =
(aq ;q) (q ;q) x q
k=0
rk
as
n → ∞,
k
Now, consider the Lebesgue measure space X, M, m, whereX, m(E) = length of E, ∀E ∈ Ω, x ∈ X, a >
∞
0, b > 0, r is an integer, s and n are positive integers, and 0 < q < 1, then {prn (x ; s, a, b ; q)}1 becomes a
∞
n
r
sequence of non negative measurable functions. Hence, {pn (xq ; s, a, b ; q)}1 will also be a sequence of non
negative measurable functions, and as n → ∞ , prn (xq n ; s, a, b ; q) → Vr (x ; s , a , b ; q) almost everywhere.
Hence, by using Fatou’s lemma, we get the inequality:
Z
Z
Vr (x ; s , a , b ; q) dm ≤ lim
prn (xq sn ; s, a, b ; q) dm
(18)
n→ ∞
E
E
3. Conclusion
The results obtained in this paper are seems to be new and quite interesting for the scope of further research. These
results give the connection between Measure theory and Special Functions in particular q-analogue.
Acknowledgement
Thanks are due to referees for their valuable suggestions and comments.
References
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[2]
[3]
[4]
[5]
[6]
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G. Gaspar and M. Rahman, Basic Hypergeometric series, Cambridge Univ. Press, 1990.
W. Hahn, Über Orthgonalpolynome, die q-differenzengleichunben genügen, Math. Nachr., 2(1949), 4-34.
D. S. Moak. The q-analogue of the Laguerre polynomials. J. Math. Anal. Appl., 81 (1981), 20-47.
H. L. Royden. Real Analysis. Macmillan Publication Co., 2002.
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