Abstract
AbstractIn this paper, we introduce the polynomials $$B^{(k)}_{n,\alpha }(x;q)$$
B
n
,
α
(
k
)
(
x
;
q
)
generated by a function including Jackson q-Bessel functions $$J^{(k)}_{\alpha }(x;q)$$
J
α
(
k
)
(
x
;
q
)
$$ (k=1,2,3),\,\alpha >-1$$
(
k
=
1
,
2
,
3
)
,
α
>
-
1
. The cases $$\alpha =\pm \frac{1}{2}$$
α
=
±
1
2
are the q-analogs of Bernoulli and Euler$$^{,}$$
,
s polynomials introduced by Ismail and Mansour for $$(k=1,2)$$
(
k
=
1
,
2
)
, Mansour and Al-Towalib for $$(k=3)$$
(
k
=
3
)
. We study the main properties of these polynomials, their large n degree asymptotics and give their connection coefficients with the q-Laguerre polynomials and little q-Legendre polynomials.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Mathematics (miscellaneous)
Reference22 articles.
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2. Ahlfors, L.: Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable. McGraw-Hill, New York (1953)
3. Al-Salam, W.A.: $$ q$$-Bernoulli numbers and polynomials. Math. Nachr. 17, 239–260 (1959)
4. Annaby, M.H., Mansour, Z.S.: On the zeros of the second and third Jackson $$q$$-Bessel functions and their associated $$q$$-Hankel transforms. Math. Proc. Camb. Philos. Soc. 147, 47–67 (2009)
5. Lecture Notes in Mathematics;MH Annaby,2012
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