EXPLICIT FORMULAS FOR MEIXNER POLYNOMIALS
DMITRY V. KRUCHININ, YURIY V. SHABLYA
Abstract. In this work, using notions of composita and composition of
generating functions, we show an easy way to obtain explicit formulas
for some current polynomials. Particularly, we consider the Meixner
polynomials of the first and second kinds.
Keywords and phrases. composita, generating function, composition
of generating function, Bessel polynomials, Meixner polynomials.
1. Introduction
There are many authors who have studied polynomials and their properties (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]). The polynomials are applied in many
areas of mathematics, for instance, continued fractions, operator theory,
analytic functions, interpolation, approximation theory, numerical analysis,
electrostatics, statistical quantum mechanics, special functions, number theory, combinatorics, stochastic processes, sorting and data compression, and
etc.
The research area of obtaining explicit formulas for polynomials has received much attention by Srivastava [11, 12], Cenkci [13], Boyadzhiev [14],
Kruchinin [15, 16, 17].
The main purpose of this paper is to obtain explicit formulas for the
Meixner polynomials of the first and second kinds.
In this paper we use a method based on a notion of composita, which
were presented in [18].
P
f (n)tn is the generating function, in
Definition 1.1. Suppose F (t) =
n>0
which there is no free term f (0) = 0. From this generating function we can
write the following condition
X
[F (t)]k =
F (n, k)tn .
n>0
The expression F (n, k) is the composita [22] and it is denoted by F ∆ (n, k).
Below we show some required rules and operations with compositae.
P
Theorem 1.1. Suppose F (t) =
f (n)tn is the generating function, and
n>0
F ∆ (n, k) is the composita of F (t), and α is constant. For the generating
function A(t) = αF (t) the composita is equal to
A∆ (n, k) = αk F ∆ (n, k).
P
Theorem 1.2. Suppose F (t) =
f (n)tn is the generating function, and
(1)
n>0
F ∆ (n, k) is the composita of F (t), and α is constant. For the generating
1
2
DMITRY V. KRUCHININ, YURIY V. SHABLYA
function A(t) = F (αt) the composita is equal to
A∆ (n, k) = αn F ∆ (n, k).
P
P
Theorem 1.3. Suppose F (t) =
f (n)tn , R(t) =
r(n)tn are generat-
(2)
n>0
n>0
ing functions, and F ∆ (n, k) , R∆ (n, k) are their compositae. Then for the
composition of generating functions A(t) = R(F (t)) the composita is equal
to
n
X
(3)
A∆ (n, k) =
F ∆ (n, m)G∆ (m, k).
m=k
P
Theorem 1.4. Suppose F (t) =
f (n)tn , R(t) =
n>0
P
r(n)tn are generating
n>0
functions, and F ∆ (n, k) is the composita of F (t). Then for the composition
of generating
functions A(t) = R(F (t)) coefficients of generating functions
P
A(t) =
a(n)tn are
n≥0
(4)
a(n) =
n
X
F ∆ (n, k) r(k),
k=1
a(0) = r(0)
P
P
r(n)tn are generating
Theorem 1.5. Suppose F (t) =
f (n)tn , R(t) =
n>0
n>0
functions. Then for the product of generating
A(t) = F (t) R(t)
P functions
coefficients of generating functions A(t) =
a(n)tn are
n≥0
(5)
a(n) =
n
X
f (k) r(n − k)
k=0
In this paper we consider an application of this mathematical tool for
the Bessel polynomials and the Meixner polynomials of the first and second
kinds.
2. Bessel polynomials
Krall and Frink [19] considered a new class of polynomials. Since the
polynomials connected with the Bessel function, they called them the Bessel
polynomials. The explicit formula for the Bessel polynomials is
n
X
(n + k)! x k
.
yn (x) =
(n − k)! k! 2
k=0
Then Carlitz [20] defined a class of polynomials associated with the Bessel
polynomials by
pn (x) = xn yn−1 (1/x).
The pn (x) is defined by the explicit formula [21]
n−k
n
X
1
(2n − k − 1)!
(6)
pn (x) =
xk
(k − 1)! (n − k)! 2
k=1
EXPLICIT FORMULAS FOR MEIXNER POLYNOMIALS
3
and by the following generating function
∞
√
X
pn (x) n
(7)
t = ex (1− 1−2t) .
n!
n=0
Using the notion of composita, we can obtain an explicit formula (6) from
the generating function (7).
√
For the generating function C(t) = 1− 21−4 t the composita is given as
follows (see [22]):
k 2n − k − 1
∆
.
C (n, k) =
n
n−1
√
We represent ex (1− 1−2t) as the composition of generating functions A(B(t)),
where
√
A(t) = x (1 − 1 − 2 t) = 2 x C(t/2),
∞
X
1 n
t
t .
B(t) = e =
n!
n=0
Then, using rules (1) and (2), we obtain the composita of A(t)
n
∆
k 1
∆
k−n k k 2 n − k − 1
A (n, k) = (2x)
C (n, k) = 2
x
.
2
n
n−1
∞
P
Coefficients of the generating function B(t) =
b(n) tn are equal to
n=0
1
.
n!
Then, using (4), we obtain the explicit formula
n
n
X
X
(2n − k − 1)! k−n k
pn (x) = n!
A∆ (n, k) b(k) =
2
x ,
(k − 1)! (n − k)!
b(n) =
k=1
k=1
which coincides with the explicit formula (6).
3. Meixner polynomials of the first kind
The Meixner polynomials of the first kind are defined by the following
recurrence relation [23, 24]:
(8)
mn+1 (x; β, c) =
(c − 1)n
((x − bn )mn (x; β, c) − λn mn−1 (x; β, c)) ,
cn
where
m0 (x; β, c) = 1,
m1 (x; β, c) = x − b0 ,
(1 + c)n + βc
bn =
,
1−c
cn(n + β − 1)
λn =
.
(1 − c)2
Using (8), we obtain the first few Meixner polynomials of the first kind
m0 (x; β, c) = 1;
m1 (x; β, c) =
x(c − 1) + βc
;
c
4
DMITRY V. KRUCHININ, YURIY V. SHABLYA
x2 (c − 1)2 + x((2β + 1)c2 − 2βc − 1) + (β + 1)βc2
.
c2
The Meixner polynomials of the first kind are defined by the following
generating function [21]:
∞
X
mn (x; β, c) n
t x
(1 − t)−x−β .
(9)
t = 1−
n!
c
m2 (x; β, c) =
n=0
Using the notion of composita, we can obtain an explicit formula mn (x; β, c)
from the generating function (9).
First, we represent the generating function (9) as a product of generating
functions A(t) B(t), where the functions A(t) and B(t) are expanded by
binomial theorem
∞ t x X x
A(t) = 1 −
=
(−1)n c−n tn ,
c
n
n=0
∞ X
−x − β
−x−β
B(t) = (1 − t)
=
(−1)n tn .
n
n=0
Coefficients of the generating functions A(t) =
∞
P
∞
P
a(n) tn and B(t) =
n=0
b(n) tn are, respectively, given as follows:
n=0
x
a(n) =
(−1)n c−n
n
and
−x − β
(−1)n .
n
Then, using (5), we obtain a new explicit formula for the Meixner polynomials of the first kind:
n n
X
X
x −x − β −k
n
a(k) b(n − k) = (−1) n!
c .
mn (x; β, c) = n!
k
n−k
b(n) =
k=0
k=0
4. Meixner polynomials of the second kind
The Meixner polynomials of the second kind are defined by the following
recurrence relation [23, 24]:
(10)
Mn+1 (x; δ, η) = (x − bn )Mn (x; δ, η) − λn Mn−1 (x; δ, η),
where
M0 (x; δ, η) = 1,
M1 (x; δ, η) = x − b0 ,
bn = (2n + η)δ,
λn = (δ 2 + 1)n(n + η − 1).
Using (10), we get the first few Meixner polynomials of the second kind
M0 (x; δ, η) = 1;
M1 (x; δ, η) = x − δη;
M2 (x; δ, η) = x2 − x2δ(1 + η) + η((η + 1)δ 2 − 1).
EXPLICIT FORMULAS FOR MEIXNER POLYNOMIALS
5
The Meixner polynomials of the second kind are defined by the following
generating function [21]:
∞
X
Mn (x; δ, η) n
t
2
2 −η/2
−1
(11)
.
t = (1 + δt) + t
exp x tan
n!
1 + δt
n=0
Using the notion of composita, we can obtain an explicit formula Mn (x; δ, η)
from the generating function (11).
We represent the generating function (11) as a product of generating
functions A(t) B(t), where
−η/2
−η/2
A(t) = (1 + δt)2 + t2
= 1 + 2δt + (δ 2 + 1)t2
,
t
.
B(t) = exp x tan−1
1 + δt
Next we represent A(t) as a composition of generating functions A1 (A2 (t))
and we expand A1 (t) by binomial theorem:
∞ X
−η/2 n
−η/2
A1 (t) = (1 + t)
=
t
n
n=0
and
A2 (t) = 2δt + (δ 2 + 1)t2 .
∞
P
a1 (n) tn are
Coefficients of generating function A1 (t) =
n=0
−η/2
a1 (n) =
.
n
The composita for the generating function ax + bx2 is given as follows
(see [15]):
k
2k−n n−k
a
b
.
n−k
Then the composita of the generating function A2 (t) equals
k
∆
2k−n 2
n−k
A2 (n, k) = (2δ)
(δ + 1)
.
n−k
Using (4), we obtain coefficients of the generating function A(t) =
∞
P
n=0
a(0) = a1 (0),
a(n) =
n
X
A∆
2 (n, k) a1 (k).
k=1
Therefore, we get the following expression
a(0) = 1,
a(n) =
n
X
k=1
(2δ)
2k−n
2
(δ + 1)
n−k
k
n−k
−η/2
.
k
a(n) tn :
6
DMITRY V. KRUCHININ, YURIY V. SHABLYA
Next we represent B(t) as a composition of generating functions B1 (B2 (t)),
where
∞
X
1 n
t ,
B1 (t) = et =
n!
n=0
t
−1
B2 (t) = x tan
.
1 + δt
We also represent B2 (t) as a composition of generating functions x C1 (C2 (t)),
where
C1 (t) = tan−1 (t),
t
C2 (t) =
.
1 + δt
The composita for the generating function tan−1 (t) is given as follows (see
[22]):
n
k! X
3n+k
n−k
2j n − 1 j
2
2
,
(−1)
+ (−1)
j! j − 1 k
2k+1
j=k
where kj is the Stirling number of the first kind.
Then the composita of generating function C1 (t) equals
n
k! X
3n+k
n−k
2j n − 1 j
∆
C1 (n, k) = (−1) 2 + (−1) 2
.
j! j − 1 k
2k+1
j=k
at
is given as follows (see
The composita for the generating function 1−bt
[22]):
n − 1 k n−k
a b
.
k−1
Therefore, the composita of generating function C2 (t) is equal to
n−1
∆
C2 (n, k) =
(−δ)n−k .
k−1
Using rules (3) and (1), we obtain the composita of generating function
B2 (t):
n
X
C2∆ (n, m)C1∆ (m, k).
B2∆ (n, k) = xk
m=k
Coefficients of generating function B1 (t) =
∞
P
b1 (n) tn are defined by
n=0
1
.
n!
Then, using rule (4), we obtain coefficients of generating function B(t) =
∞
P
b(n) tn :
b1 (n) =
n=0
b(0) = b1 (0),
n
X
b(n) =
B2∆ (n, k) b1 (k).
k=1
After some transformations, we obtain the following expression
b(0) = 1,
EXPLICIT FORMULAS FOR MEIXNER POLYNOMIALS
n
n X
xk X n n−m
b(n) =
δ
n
m
k=1
m=k
1 + (−1)m+k
(−1)
m+k
−n
2
!
7
m
X
2j−k−1 m j
.
(j − 1)! j
k
j=k
Therefore, using (5), we obtain a new explicit formula for the Meixner
polynomials of the second kind
n
X
Mn (x; δ, η) = n!
a(k) b(n − k).
k=0
References
[1] R. P. Jr. Boas and R. C. Buck, Polynomial Expansions of Analytic Functions, SpringerVerlag, (1964).
[2] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted
Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, (1984).
[3] H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and
Integrals, Elsevier Science Publishers, Amsterdam, London and New York, (2012).
[4] H. M. Srivastava, Some generalizations and basic (or q-) extensions of the Bernoulli,
Euler and Genocchi polynomials, Applied Mathematics and Information Sciences 5,
390–444 (2011).
[5] Y. Simsek and M. Acikgoz, A new generating function of (q-) Bernstein-type polynomials and their interpolation function, Abstr. Appl. Anal., (2010).
[6] H. Ozden, Y. Simsek and H. M. Srivastava, A unified presentation of the generating
functions of the generalized Bernoulli, Euler and Genocchi polynomials, Computers
Math. Applic. 60, (2010), 2779–2787.
[7] R. Dere and Y. Simsek, Applications of umbral algebra to some special polynomials,
Adv. Studies Contemp. Math. 22, (2012), 433–438.
[8] Y. Simsek, Complete sum of products of (h,q)-extension of Euler polynomials and numbers, Journal of Difference Equations and Applications 16(11), (2010), 1331–1348.
[9] Y. He and S. Araci, Sums of products of Apostol-Bernoulli and Apostol-Euler polynomials, Advances in Difference Equations 2014: 155.
[10] S. Araci, Novel identities involving Genocchi numbers and polynomials arising from
applications of umbral calculus, Applied Mathematics and Computation 233, (2014),
599–607.
[11] H. M. Srivastava and Guo-Dong Liu, Explicit formulas for the Norlund polynomials
(x)
(x)
Bn and bn , Computers Math. Applic. 51, (2006), 1377–1384.
[12] H. M. Srivastava and P. G. Todorov, An explicit formula for the generalized Bernoulli
polynomials, J. Math. Anal. Appl. 130, (1988), 509–513.
[13] M. Cenkci, An explicit formula for generalized potential polynomials and its applications, Discrete Mathematics 309, (2009), 1498–1510.
[14] K. N. Boyadzhiev, Derivative polynomials for Tanh, Tan, Sech and Sec in explicit
form, Fibonacci Quarterly 45(4), (2007), 291–303.
[15] D. V. Kruchinin and V. V. Kruchinin, Application of a composition of generating
functions for obtaining explicit formulas of polynomials, J. Math. Anal. Appl. 404,
(2013), 161–171.
[16] D. V. Kruchinin and V. V. Kruchinin, Explicit formulas for some generalized polynomials, Appl. Math. Inf. Sci. 7(5), (2013), 2083–2088.
[17] D. V. Kruchinin, Explicit formula for generalized Mott polynomials, Advanced Studies
in Contemporary Mathematics 24(3), (2014), 327–322.
[18] D. V. Kruchinin and V. V. Kruchinin, A method for obtaining expressions for polynomials based on a composition of generating functions, Numerical Analysis and Applied
Mathematics ICNAAM 2012, AIP Conf. Proc. 1479, (2012), 383–386.
[19] H. L. Krall and O. Frink, A new class of ortogonal polynomials: The Bessel polynomials, Trans. Amer. Math. Soc. 65, (1949), 100–115.
[20] L. Carlitz, A note on the Bessel polynomials, Duke Math. J. 24, (1957), 151–162.
[21] S. Roman, The Umbral Calculus, Academic Press, (1984).
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DMITRY V. KRUCHININ, YURIY V. SHABLYA
[22] V. V. Kruchinin and D. V. Kruchinin Composita and its properties, Journal of Analysis and Number Theory 2 (2), (2014), 37–44.
[23] G. Hetyei Meixner polynomials of the second kind and quantum algebras representing
su(1,1), Proceedings of the Royal Society A: Mathematical, Physical and Engineering
Sciences 466(2117), (2010), 1409–1428.
[24] T. S. Chihara An Introduction to Orthogonal Polynomials, Gordon and Breach,
(1978).
Tomsk State University of Control Systems and Radioelectronics, Tomsk,
Russia
National Research Tomsk Polytechnical University, Tomsk, Russia
E-mail address: [email protected]