Hindawi Publishing Corporation International Journal of Combinatorics Volume 2013, Article ID 528584, 6 pages http://dx.doi.org/10.1155/2013/528584
Research Article Some Inverse Relations Determined by Catalan Matrices Sheng-liang Yang Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China Correspondence should be addressed to Sheng-liang Yang;
[email protected] Received 24 June 2013; Revised 21 August 2013; Accepted 21 August 2013 Academic Editor: Laszlo A. Szekely Copyright Β© 2013 Sheng-liang Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We use the A-sequence and Z-sequence of Riordan array to characterize the inverse relation associated with the Riordan array. We apply this result to prove some combinatorial identities involving Catalan matrices and binomial coefficients. Some matrix identities obtained by Shapiro and Radoux are all special cases of our identity. In addition, a unified form of Catalan matrices is introduced.
1. Introduction The Catalan numbers πΆπ have been widely encountered and investigated [1, 2]. They can be defined through binomial coefficients 1 2π ( ) , for π β₯ 0, πΆπ = (1) π+1 π or by the generating function πΆ(π‘) =
ββ π=0
π
πΆπ π‘ being
1 β β1 β 4π‘ πΆ (π‘) = , 2π‘
(2)
which satisfies the functional equation πΆ(π‘) = 1 + π‘πΆ(π‘)2 . In [2], Stanley listed 66 enumerative problems which are counted by the Catalan numbers. Many number triangles related to the Catalan sequence have been introduced in the literature. In [3β5], Shapiro et al. introduced a Catalan triangle π΅ with the entries given by π΅π,π =
π + 1 2π + 2 ( ), π+1 πβπ
where π β₯ π β₯ 0.
(3)
The following identity is obtained in [6] in connection with the moment of the Catalan triangle: 1 2 (5 (14 42 .. (.
0 1 4 14 48 .. .
0 0 1 6 27 .. .
0 0 0 1 8 .. .
0 0 0 0 1 .. .
1 β
β
β
1 4 β
β
β
2 4 β
β
β
) (3) ( 2 ) β
β
β
) (4) = (43 ) . β
β
β
5 44 . . d) ( .. ) ( .. )
Another proof of the above identity is given by Woan et al. [7] while computing the areas of parallelo-polyominoes via generating functions. In [8], a combinatorial interpretation of the matrix identity (4) is also obtained. In [9], Radoux introduced a triangle of numbers ππ,π =
2π + 1 2π ( ), π+π+1 πβπ
(5)
and he presents the identity βππ=0 (2π + 1)ππ,π = 22π with π β₯ π β₯ 0, which is equivalent to following matrix equation: 1 1 (2 (5 14 .. (.
0 1 3 9 28 .. .
0 0 1 5 20 .. .
0 0 0 1 7 .. .
0 0 0 0 1 .. .
1 β
β
β
1 4 β
β
β
3 β
β
β
) (5) (42 ) β
β
β
) (7) = (43 ) . β
β
β
9 44 . . d) ( .. ) ( .. )
(6)
Deng and Yan [10] proved this identity by using the Riordan array method. Aigner [11] introduced a number triangle with the entries given by ππ,π =
(4)
where π β₯ π β₯ 0,
π + 1 2π β π ( ), π+1 πβπ
where π β₯ π β₯ 0.
(7)
This array is also discussed in [12β14]. We use the π΄-sequence and π-sequence of Riordan array to characterize the inverse relation associated with the
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Riordan array. We apply this result to prove some combinatorial identities involving Catalan matrices and binomial coefficients, which are generalizations of (4) and (6). In addition, a unified form of Catalan matrices is introduced.
Lemma 2 (see [22, 23]). Let π· = (π(π‘), π(π‘)) be a Riordan array, and let π΄(π‘) and π(π‘) be the generating functions for the corresponding π΄- and π-sequences, respectively. Then we have π (π‘) =
2. Riordan Arrays In the recent literature, one may find that Riordan arrays have attracted the attention of various authors from many points of view, and many examples and applications can be found (see, e.g., [13, 15β21]). An infinite lower triangular matrix π· = (ππ,π )π,πβ₯0 is called a Riordan array if its column π has generating function π(π‘)π(π‘)π , where π(π‘) and π(π‘) are formal power series with π0 = 1, π0 = 0, and π1 =ΜΈ 0. The Riordan array is denoted by π· = (π(π‘), π(π‘)). Thus, the general term of Riordan array π· = (π(π‘), π(π‘)) is given by ππ,π = [π‘π ] π (π‘) π(π‘)π ,
(8)
where [π‘π ]β(π‘) denotes the coefficient of π‘π in power series β(π‘). Suppose we multiply the array π· = (π(π‘), π(π‘)) by a column vector (π0 , π1 , π2 , . . . )π and get a column vector (π0 , π1 , π2 , . . . )π . Let π(π‘) be the ordinary generating function for the sequence (π0 , π1 , π2 , . . . )π . Then it follows that the ordinary generating function for the sequence (π0 , π1 , π2 , . . . )π is π(π‘)π(π(π‘)). If we identify a sequence with its ordinary generating function, the composition rule can be rewritten as (π (π‘) , π (π‘)) π (π‘) = π (π‘) π (π (π‘)) .
(9)
This is called the fundamental theorem for Riordan arrays, and this leads to the multiplication rule for the Riordan arrays: (π (π‘) , π (π‘)) (β (π‘) , π (π‘)) = (π (π‘) β (π (π‘)) , π (π (π‘))) . (10) The set of all Riordan arrays forms a group under ordinary multiplication. The identity is (1, π‘). The inverse of (π(π‘), π(π‘)) is β1
(π (π‘) , π (π‘))
=(
1 π (π (π‘))
, π (π‘)) ,
(11)
where π(π‘) is compositional inverse of π(π‘). Lemma 1 (see [22, 23]). Let π· = (ππ,π ) be an infinite lower triangular matrix. Then π· is a Riordan array if and only if π0,0 = 1 and there exist two sequences π΄ = (ππ )πβ₯0 and π = (π§π )πβ₯0 with π0 =ΜΈ 0 and π§0 =ΜΈ 0 such that ππ+1,π+1 = π0 ππ,π + π1 ππ,π+1 + π2 ππ,π+2 + . . . ,
π, π = 0, 1, . . . ,
ππ+1,0 = π§0 ππ,0 + π§1 ππ,1 + π§2 ππ,2 + . . . ,
π = 0, 1, . . . . (12)
Such sequences are called the π΄-sequence and the π-sequence of the Riordan array π· = (π(π‘), π(π‘)), respectively.
1 , 1 β π‘π (π (π‘))
π (π‘) = π‘π΄ (π (π‘)) .
(13)
If the inverse of π· = (π(π‘), π(π‘)) is π·β1 = (π(π‘), β(π‘)). Then π΄ (π‘) =
π‘ , β (π‘)
π (π‘) =
1 β π (π‘) . β (π‘)
(14)
Example 3. (a) It is well known that the Pascal matrix π = (( ππ ))π,πβ₯0 can be expressed as the Riordan array (1/(1 β π‘), π‘/(1 β π‘)), and the generating functions of its π΄- and π-sequences are π΄(π‘) = 1 + π‘, π(π‘) = 1. More generally, it is easy to show that the generalized Pascal array π[π₯] = (π₯πβπ ( ππ ))π,πβ₯0 can be expressed as the Riordan matrix (1/(1 β π₯π‘), π‘/(1 β π₯π‘)) and the generating functions of its π΄- and π-sequences are π΄(π‘) = 1 + π₯π‘, and π(π‘) = π₯. (b) For nonnegative integer π , the Pascal functional π -eliminated matrix ππ was introduced in [24] by ππ = π+π ))π,πβ₯0 . We have ππ = (1/(1 β π₯π‘)π +1 , π‘/(1 β π₯π‘)), (π₯πβπ ( π+π and the generating functions of its π΄- and π-sequences are π΄(π‘) = 1+π₯π‘, π(π‘) = ((1+π₯π‘)π +1 β1)/(π‘(1+π₯π‘)π ). We also have β1 ππ β1 = (1/(1 β π₯π‘)π +1 , π‘/(1 β π₯π‘)) = (1/(1+π₯π‘)π +1 , π‘/(1+π₯π‘)). Definition 4. Let (ππ (π₯))πβ₯0 be a sequence of polynomials, where ππ (π₯) is of degree π and ππ (π₯) = βππ=0 ππ,π π₯π . We say that (ππ (π₯))πβ₯0 is a polynomial sequence of Riordan type if the coefficient matrix (ππ,π )π,πβ₯0 is an element of the Riordan group; that is, there exists a Riordan array (π(π‘), π(π‘)) such that (ππ,π )π,πβ₯0 = (π(π‘), π(π‘)). In this case, we say that (ππ (π₯))πβ₯0 is the polynomial sequence associated to the Riordan array (π(π‘), π(π‘)). If (ππ (π₯))πβ₯0 is the polynomial sequence associated to a π Riordan array (π(π‘), π(π‘)), and let π(π‘, π₯) = ββ π=0 ππ (π₯)π‘ be its generating function, then by (9), we have (π (π‘) , π (π‘))
1 = π (π‘, π₯) . 1 β π₯π‘
(15)
Thus, π(π‘, π₯) = π(π‘)/(1 β π₯π(π‘)). The notion of the polynomial sequence of Riordan type was introduced in [25], and it has been studied by [26]. In this paper, by studying the polynomial sequence of Riordan type related to some Catalan type matrices, we obtain some interesting identities and inverse relations. Theorem 5. Let π· = (π(π‘), π(π‘)) be a Riordan array, and let π΄(π‘) and π(π‘) be the generating functions of its π΄-sequence and π-sequence. If π΅(π‘) = (π΄(π‘) β π‘π(π‘))/(π΄(π‘) β π₯π‘), then (π (π‘) , π (π‘)) π΅ (π‘) = where π₯ is any real number.
1 , 1 β π₯π‘
(16)
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Proof. Let π΅(π‘) = (π΄(π‘) β π‘π(π‘))/(π΄(π‘) β π₯π‘); then (π(π‘), π(π‘))π΅(π‘) = π(π‘)π΅(π(π‘)) = (1/(1 β π‘π(π(π‘))))((π΄(π(π‘)) β π(π‘)π(π(π‘)))/(π΄(π(π‘)) β π₯π(π‘))) = (1/(1 β π‘π(π(π‘))))((π(π‘) β π‘π(π‘)π(π(π‘)))/(π(π‘) β π₯π(π‘))) = 1/(1 β π₯π‘). Corollary 6. Let π· = (π(π‘), π(π‘)) be a Riordan array. If π΄(π‘) and π(π‘) are the generating functions of its π΄sequence and π-sequence, respectively, and if (ππ (π₯))πβ₯0 is the polynomial sequence associated to the Riordan array π·β1 = (1/π(π(π‘)), π(π‘)), then β
β ππ (π₯) π‘π =
π=0
π΄ (π‘) β π‘π (π‘) . π΄ (π‘) β π₯π‘
(17)
The generating functions of its π΄- and π-sequence are π΄(π‘) = 1 + 2π‘ + π‘2 , π(π‘) = 2 + π‘. Because (π΄(π‘) β π‘π(π‘))/(π΄(π‘) β 4π‘) = π 1/(1 β π‘)2 = ββ π=0 (π + 1)π‘ , from Theorem 5, we have (πΆ(π‘)2 , π‘πΆ(π‘)2 )
1 π΄ (π‘) β π‘π (π‘) = . π΄ (π‘) β π₯π‘ 1 β π₯π‘
(18)
Hence, (π (π‘) , π (π‘))
β1
1 π΄ (π‘) β π‘π (π‘) = . 1 β π₯π‘ π΄ (π‘) β π₯π‘
(19)
Example 9. The lower triangular matrix in (6) can be written as (πΆ (π‘) , π‘πΆ(π‘)2 ) = (
π
β ππ,π ππ (π₯) = π₯π ,
π=0 π
(20) π
1 β β1 β 4π‘ 1 β 2π‘ β β1 β 4π‘ , ). 2π‘ 2π‘ (24)
The generating functions of its π΄- and π-sequence are π΄(π‘) = (1 + π‘)2 , π(π‘) = 1 + π‘. Since (π΄(π‘) β π‘π(π‘))/(π΄(π‘) β 4π‘) = π (1 + π‘)/(1 β π‘)2 = ββ π=0 (2π + 1)π‘ , from Theorem 5, we have (πΆ (π‘) , π‘πΆ(π‘)2 )
The result then follows from identity (15). Corollary 7. Let π· = (ππ,π ) be a Riordan array. If π΄(π‘) and π(π‘) are the generating functions of its π΄-sequence and π-sequence, respectively, and if (ππ (π₯))πβ₯0 is the polynomial sequence associated to the Riordan array π·β1 , then
(23)
This is exactly the matrix identity (4).
Proof. By the theorem, we have (π (π‘) , π (π‘))
1 1 . = 2 1 β 4π‘ (1 β π‘)
1+π‘ 1 = . (1 β π‘)2 1 β 4π‘
(25)
This is exactly the matrix identity (6).
3. Inverse Relations Determined by Catalan Matrices Let π, π be integer numbers, and let π be arbitrary parameter. We define the generalized Catalan matrix πΆ[π, π; π] to be the Riordan array
β ππ,π π₯ = ππ (π₯) ,
πΆ [π, π; π] = (πΆ(ππ‘)π , π‘πΆ(ππ‘)π ) ,
where ππ,π is the (π, π)-element of π·β1 . In matrix form, we have
where πΆ(π‘) is the generating function of Catalan sequence defined in (2). From [27], we have πΆ(π‘)π = ββ π=0 π (π/(2π + π)) ( 2π+π π ) π‘ for any integer number π. Hence [π‘π ]πΆ(ππ‘)π (π‘πΆ(ππ‘)π )π = [π‘πβπ ]πΆ(ππ‘)π+ππ = ((π + ππ)/(2π β ) ππβπ . Therefore, by (8), the generic 2π + π + ππ)) ( 2πβ2π+π+ππ πβπ element of the generalized Catalan matrix πΆ[π, π; π] is given by
π=0
π0,0 π1,0 (π2,0 (π3,0 π4,0 .. ( .
0 0 0 0 π1,1 0 0 0 π2,1 π2,2 0 0 π3,1 π3,2 π3,3 0 π4,1 π4,2 π4,3 π4,4 .. .. .. .. . . . . 1 π₯ 2 (π₯ ) = (π₯ 3 ) . π₯4 .. (.)
β
β
β
π0 (π₯) π1 (π₯) β
β
β
β
β
β
) (π2 (π₯)) β
β
β
) (π3 (π₯)) π4 (π₯) β
β
β
. d) ( .. )
(21)
Example 8. The lower triangular matrix in (4) may be represented as (πΆ(π‘)2 , π‘πΆ(π‘)2 ) = (
1 β 2π‘ β β1 β 4π‘ 1 β 2π‘ β β1 β 4π‘ , ). 2π‘2 2π‘ (22)
πΆ[π, π; π]π,π =
(26)
π + ππ 2π β 2π + π + ππ πβπ ( )π . πβπ 2π β 2π + π + ππ (27)
Denote πΆ[π, π] = πΆ[π, π; 1] = (πΆ(π‘)π , π‘πΆ(π‘)π ). For 0 β€ π, π β€ 2, the corresponding matrices πΆ[π, π] are widely studied by many authors [5, 12, 14, 15, 28, 29]. For example, πΆ[1, 0] = (πΆ(π‘), π‘) = ((1 β β1 β 4π‘)/2π‘, π‘), πΆ[0, 1] = (1, π‘πΆ(π‘)), πΆ[2, 0] = (πΆ(π‘)2 , π‘), and πΆ[0, 2] = (1, π‘πΆ(π‘)2 ) = (1, πΆ(π‘) β 1). The matrix πΆ[1, 1] = (πΆ(π‘), π‘πΆ(π‘)) is the Catalan triangle introduced by Aigner [11] and studied in [6, 12, 13]. The matrix πΆ[2, 2] = (πΆ(π‘)2 , π‘πΆ(π‘)2 ) is the Catalan triangle defined by Shapiro [5]; see also (4). The matrix πΆ[1, 2] = (πΆ(π‘), π‘πΆ(π‘)2 ) is the Catalan matrix defined by Radoux [9]; see also (6).
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Theorem 10. Let {πΉπ (π₯, π)} be the polynomial sequence associated to the Riordan array πΆ[1, 1; π]β1 = (πΆ(ππ‘), π‘πΆ(ππ‘))β1 . Then, the identities π
π + 1 2π β π πβπ ( ) π πΉπ (π₯, π) = π₯π , πβπ π + 1 π=0 β
π
π+1 π ) π₯ (βπ)πβπ = πΉπ (π₯, π) , β( πβπ
π
(28)
π=0
hold for every π β N. Proof. From generic term given in (27) with π = 1 and π = 1, we have the generic term of Catalan matrix πΆ[1, 1; π] = (πΆ(ππ‘), π‘πΆ(ππ‘)) which is πΆ[1, 1; π]π,π = ((1 + π)/(2π β 2π + 1 + π)) ( 2πβ2π+1+π ) ππβπ , and by simplifying we obtain πβπ πβπ πΆ[1, 1; π]π,π = ((π + 1)/(π + 1)) ( 2πβπ . Using (11) and (14), πβπ ) π β1 we get πΆ[1, 1; π] = (1 β ππ‘, π‘(1 β ππ‘)), and the generating functions of its π΄- and π-sequences are π΄(π‘) = 1/(1 β ππ‘) π = and π(π‘) = π/(1 β ππ‘). By Corollary 6, ββ π=0 πΉπ (π₯, π)π‘ 2 (π΄(π‘) β π‘π(π‘))/(π΄(π‘) β π₯π‘) = (1 β ππ‘)/(1 β π₯π‘ + π₯ππ‘ ). Therefore, πΉ0 (π₯, π) = 1, πΉ1 (π₯, π) = π₯ β π, and πΉπ (π₯, π) = π₯πΉπβ1 (π₯, π) β π₯ππΉπβ2 (π₯, π), for π β₯ 2. Solving this recurrence relation, we π+1 ) π₯π (βπ)πβπ . From Corollary 7, we have πΉπ (π₯, π) = βππ=0 ( πβπ get the results. π For the case π = 1, π₯ = 4, we have ββ π=0 πΉπ (4, 1)π‘ = (1 β β 2 2 πβ1 π‘)/(1 β 4π‘ + 4π‘ ) = (1 β π‘)/(1 β 2π‘) = βπ=0 2 (π + 2)π‘π . Thus, πΉπ (4, 1) = 2πβ1 (π + 2). By Theorem 10, we obtain π
π
π + 1 2π β π πβ1 ( ) 2 (π + 2) = 4π . β π π + 1 π=0
0 1 2 5 14 42 .. .
0 0 1 3 9 28 .. .
0 0 0 1 4 14 .. .
0 0 0 0 1 5 .. .
0 0 0 0 0 1 .. .
(29)
Theorem 11. Let {π’π (π₯, π)} be the polynomial sequence associated to the Riordan array πΆ[1, 2; π]β1 = (πΆ(ππ‘), π‘πΆ(ππ‘)2 )β1 . Then, for any nonnegative integer π, one has π
β
π
π+π π ) π₯ (βπ)πβπ = π’π (π₯, π) . β( πβπ
π=0
π
π+π π ) 4 = 2π + 1, β (β1)πβπ ( πβπ
π=0
(32)
π
2π + 1 2π + 1 ( ) (2π + 1) = 4π . β πβπ 2π + 1 π=0
The last identity is equivalent to identity (6). For the case π = β1 and π₯ = 1, we have βππ=0 π’π (1, β1)π‘π = (1βπ‘)/(1β3π‘+π‘2 ) = βππ=0 πΉ2π+1 π‘π , and π’π (1, β1) = πΉ2π+1 , where π {πΉπ }πβ₯0 is Fibonacci sequence with generating ββ π=0 πΉπ π‘ = 2 π‘/(1 β π‘ β π‘ ). By Theorem 11, we have
π=0
2π + 1 2π + 1 ( ) πΉ2π+1 = 1, 2π + 1 π β π (33)
π
π+π π ) 4 = πΉ2π+1 . β( πβπ
π=0
1 β
β
β
1 4 β
β
β
3 2 β
β
β
) ( 8 ) (4 ) ( ) ( 3) β
β
β
) ) ( 20 ) = (44 ) . β
β
β
) ( 48 ) (4 ) β
β
β
112 45 .. . d) ( . ) ( .. ) (30)
2π + 1 2π + 1 πβπ ( ) π π’π (π₯, π) = π₯π , πβπ 2π + 1 π=0
Since πΆ[1, 2; π]β1 = (1/(1+ππ‘), π‘/(1+ππ‘)2 ), from Lemma 2, the generating functions of π΄- and π-sequences of πΆ[1, 2; π] are π΄(π‘) = (1+ππ‘)2 and π(π‘) = π+π2 π‘. Hence, βππ=0 π’π (π₯, π)π‘π = (π΄(π‘) β π‘π(π‘))/(π΄(π‘) β π₯π‘) = (1 + ππ‘)/(1 β (π₯ β 2π)π‘ + π2 π‘2 ). For the case π = 1 and π₯ = 4, we have βππ=0 π’π (4, 1)π‘π = (1+π‘)/(1β 2π‘+π‘2 ) = (1+π‘)/(1βπ‘)2 = βππ=0 (2π+1)π‘π , and π’π (4, 1) = 2π+1. By Theorem 11, we have
β (β1)πβπ
The last identity is equivalent to the following matrix identity: 1 1 (2 (5 ( (14 42 .. (.
[π‘π ](1/(1 + ππ‘))(π‘/(1 + ππ‘)2 ) = [π‘πβπ ](1/(1 + ππ‘))2π+1 = (βπ)πβπ ( π+π πβπ ). By Corollary 7, we obtain the desired results.
π
π+1 π ) 4 (β1)πβπ = 2πβ1 (π + 2) , β( πβπ
π=0
Proof. From generic term given in (27) with π = 1 and π = 2, we have the generic term of Catalan matrix πΆ[1, 2; π] = (πΆ(ππ‘), π‘πΆ(ππ‘)2 ) which is πΆ[1, 2; π]π,π = ((2π + 1)/(2π + πβπ 1)) ( 2π+1 . Using (11), we obtain πΆ[1, 2; π]β1 = (1/(1 + πβπ ) π 2 ππ‘), π‘/(1 + ππ‘) ). Hence, the generic term of πΆ[1, 2; π]β1 is
(31)
Theorem 12. Let {Vπ (π₯, π)} be the polynomial sequence associated to the Riordan array πΆ[2, 2; π]β1 = (πΆ(ππ‘)2 , π‘πΆ(ππ‘)2 )β1 . Then, one has π
π + 1 2π + 2 πβπ ( ) π Vπ (π₯, π) = π₯π , πβπ π + 1 π=0 β
(34)
π
π+π+1 π ) π₯ (βπ)πβπ = Vπ (π₯, π) , β( πβπ
π=0
for every π β N. Proof. From generic term given in (27) with π = 2 and π = 2, we have the generic term of Catalan matrix πΆ[2, 2; π] = (πΆ(ππ‘)2 , π‘πΆ(ππ‘)2 ) which is πΆ[2, 2; π]π,π = ((π + 1)/ πβπ . Using (11), we obtain πΆ[2, 2; π]β1 (π + 1)) ( 2π+2 πβπ ) π 2 = (1/(1 + ππ‘) , π‘/(1 + ππ‘)2 ). Hence, the generic term π
of πΆ[2, 2; π]β1 is [π‘π ](1/(1 + ππ‘)2 )(π‘/(1 + ππ‘)2 ) = ). By Corollary 7, [π‘πβπ ](1/(1 + ππ‘))2π+2 = (βπ)πβπ ( π+π+1 πβπ we obtain the desired results.
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The generating functions of π΄- and π-sequences of πΆ[2, 2; π] are π΄(π‘) = 1 + 2ππ‘ + π2 π‘2 and π(π‘) = 2π + π2 π‘. Thus π ββ π=0 Vπ (π₯, π)π‘ = (π΄(π‘)βπ‘π(π‘))/(π΄(π‘)βπ₯π‘) = 1/(1β(π₯β2π)π‘+ 2 2 π π‘ ). For the case π = 1 and π₯ = 2π¦ + 2, we have ββ π=0 Vπ (2π¦ + π π (π¦)π‘ , where ππ (π¦) are 2, 1)π‘π = 1/(1 β 2π¦π‘ + π‘2 ) = ββ π=0 π Chebyshev polynomials of the second kind (see [1]). Hence, by Theorem 13, we have π
π+π+1 π ) (2π¦ + 2) (β1)πβπ = ππ (π¦) , β( πβπ
π=0
π
π + 1 2π + 2 π ( ) ππ (π¦) = (2π¦ + 2) . β π β π π+1 π=0
(35)
π
β
π+2 π ) π₯ (βπ)πβπ = π€π (π₯, π) . β( πβπ
(36)
π=0
Proof. Using (11), we obtain πΆ[2, 1; π]β1 = (πΆ(ππ‘)2 , π‘πΆ(ππ‘))β1 = ((1 β ππ‘)2 , π‘ β ππ‘2 ). Hence, the generic term of πΆ[2, 2; π]β1 π π+2 ). is [π‘π ](1 β ππ‘)2 (π‘ β ππ‘2 ) = [π‘πβπ ](1 β ππ‘)π+2 = (βπ)πβπ ( πβπ From generic term given in (27) with π = 2 and π = 1, we have the generic term of Catalan matrix πΆ[2, 1; π] = (πΆ(ππ‘)2 , π‘πΆ(ππ‘)) which is πΆ[2, 1; π]π,π = ((2 + π)/(2π β 2π + 2 + π)) ( 2πβ2π+2+π ) ππβπ , and by simplifying we obtain πβπ πβπ πΆ[1, 1; π]π,π = ((π+2)/(π+2)) ( 2πβπ+1 . From Corollary 7, πβπ ) π we obtain the desired results.
The generating functions for the π΄- and π-sequences of πΆ[2, 1; π] are π΄(π‘) = 1/(1 β ππ‘), π(π‘) = (2π β π2 π‘)/(1 β ππ‘), and 2 π ββ π=0 π€π (π₯, π)π‘ = (π΄(π‘)βπ‘π(π‘))/(π΄(π‘)βπ₯π‘) = (1βππ‘) /(1βπ₯π‘+ β 2 π₯ππ‘ ). For the case π = 1 and π₯ = 4, we have βπ=0 π€π (4, 1)π‘π = πβ2 π π‘ . (1βπ‘)2 /(1β4π‘+4π‘2 ) = (1βπ‘)2 /(1β2π‘)2 = 1+ββ π=1 (π+3)2 By Theorem 13, we have π
π+2 π ) 4 = (π + 3) 2πβ2 , β (β1)πβπ ( πβπ
π β₯ 1,
(37)
π=0
π π + 2 2π β π + 1 2 2π + 1 ( ( )+β ) (π + 3) 2πβ2 π+1 π+2 π+1 π + 2 π=1 (38)
= 4π ,
π β₯ 0.
0 1 3 9 28 .. .
0 0 1 4 14 .. .
0 0 0 1 5 .. .
0 0 0 0 1 .. .
1 β
β
β
1 4 β
β
β
2 β
β
β
) ( 5 ) ( 42 ) β
β
β
) (12) = (43 ) . (39) β
β
β
28 44 . . d) ( .. ) ( .. )
The author would like to thank the referees for their helpful suggestions. This work was supported by the National Natural Science Foundation of China (11261032).
References
Theorem 13. Let {π€π (π₯, π)} be the polynomial sequence associated to the Riordan array πΆ[2, 1; π]β1 = (πΆ(ππ‘)2 , π‘πΆ(ππ‘))β1 . Then, one has identities
π
1 2 (5 (14 42 .. (.
Acknowledgments
Substituting π¦ = 1 in the last identity, we get (4) again.
π + 2 2π β π + 1 πβπ ( ) π π€π (π₯, π) = π₯π , πβπ π + 2 π=0
The matrix form of the last identity is
[1] L. Comtet, Advanced Combinatorics, D. Reidel Publishing, Dordrecht, The Netherlands, 1974. [2] R. P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, UK, 1999. [3] D. G. Rogers, βPascal triangles, Catalan numbers and renewal arrays,β Discrete Mathematics, vol. 22, no. 3, pp. 301β310, 1978. [4] C. L. Mallows and L. Shapiro, βBalls on the lawn,β Journal of Integer Sequences, vol. 2, article 99.1.5, 1999. [5] L. W. Shapiro, βA Catalan triangle,β Discrete Mathematics, vol. 14, no. 1, pp. 83β90, 1976. [6] L. W. Shapiro, W.-J. Woan, and S. Getu, βRuns, slides and moments,β SIAM Journal on Algebraic Discrete Methods, vol. 4, no. 4, pp. 459β466, 1983. [7] W.-J. Woan, L. Shapiro, and D. G. Rogers, βThe Catalan numbers, the Lebesgue integral, and 4πβ2 ,β The American Mathematical Monthly, vol. 104, no. 10, pp. 926β931, 1997. [8] Y. C. Chen, Y. Li, L. W. Shapiro, and H. F. Yan, βMatrix identities on weighted partial Motzkin paths,β European Journal of Combinatorics, vol. 28, no. 4, pp. 1196β1207, 2007. [9] C. Radoux, βAdditional formulas for polynomials built on classical combinatorial sequences,β Journal of Computational and Applied Mathematics, vol. 115, no. 1-2, pp. 471β477, 2000. [10] E. Deng and W. J. Yan, βSome identities on the Catalan, Motzkin and SchrΒ¨oder numbers,β Discrete Applied Mathematics, vol. 156, no. 14, pp. 2781β2789, 2008. [11] M. Aigner, βEnumeration via ballot numbers,β Discrete Mathematics, vol. 308, no. 12, pp. 2544β2563, 2008. [12] T. X. He, βParametric Catalan numbers and Catalan triangles,β Linear Algebra and Its Applications, vol. 438, no. 3, pp. 1467β 1484, 2013. [13] G. S. Cheon, H. Kim, and L. W. Shapiro, βCombinatorics of Riordan arrays with identical π΄ and π sequences,β Discrete Mathematics, vol. 312, no. 12-13, pp. 2040β2049, 2012. [14] S. StanimiroviΒ΄c, P. StanimiroviΒ΄c, and A. IliΒ΄c, βBallot matrix as Catalan matrix power and related identities,β Discrete Applied Mathematics, vol. 160, no. 3, pp. 344β351, 2012. [15] L. W. Shapiro, S. Getu, W.-J. Woan, and L. Woodson, βThe Riordan group,β Discrete Applied Mathematics, vol. 34, no. 1β3, pp. 229β239, 1991. [16] R. Sprugnoli, βRiordan arrays and combinatorial sums,β Discrete Mathematics, vol. 132, no. 1β3, pp. 267β290, 1994.
6 [17] P. Peart and L. Woodson, βTriple factorization of some Riordan matrices,β The Fibonacci Quarterly, vol. 31, no. 2, pp. 121β128, 1993. [18] P. Peart and W. Woan, βA divisibility property for a subgroup of Riordan matrices,β Discrete Applied Mathematics, vol. 98, no. 3, pp. 255β263, 2000. [19] P. Barry, βA note on a one-parameter family of Catalan-like numbers,β Journal of Integer Sequences, vol. 12, no. 5, article 09.5.4, 2009. [20] N. T. Cameron and A. Nkwanta, βOn some (pseudo) involutions in the Riordan group,β Journal of Integer Sequences, vol. 8, no. 3, article 05.3.7, 2005. [21] A. LuzΒ΄on, D. Merlini, M. MorΒ΄on, and R. Sprugnoli, βIdentities induced by Riordan arrays,β Linear Algebra and Its Applications, vol. 436, no. 3, pp. 631β647, 2012. [22] T. X. He and R. Sprugnoli, βSequence characterization of Riordan arrays,β Discrete Mathematics, vol. 309, no. 12, pp. 3962β 3974, 2009. [23] D. Merlini, D. G. Rogers, R. Sprugnoli, and M. C. Verri, βOn some alternative characterizations of Riordan arrays,β Canadian Journal of Mathematics, vol. 49, no. 2, pp. 301β320, 1997. [24] M. Bayat and H. Teimoori, βPascal π-eliminated functional matrix and its property,β Linear Algebra and Its Applications, vol. 308, no. 1β3, pp. 65β75, 2000. [25] S.-L Yang and S.-N. Zheng, βDeterminant representations of polynomial sequences of Riordan type,β Journal of Discrete Mathematics, vol. 2013, Article ID 734836, 6 pages, 2013. [26] A. LuzΒ΄on and M. A. MorΒ΄on, βRecurrence relations for polynomial sequences via Riordan matrices,β Linear Algebra and Its Applications, vol. 433, no. 7, pp. 1422β1446, 2010. [27] R. Graham, D. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, New York, NY, USA, 1989. [28] A. Nkwanta and E. Barnes, βTwo Catalan-type Riordan arrays and their connections to the Chebyshev polynomials of the first kind,β Journal of Integer Sequences, vol. 15, no. 3, article 12.3.3, 2012. [29] S. StanimiroviΒ΄c, P. StanimiroviΒ΄c, M. MiladinoviΒ΄c, and A. IliΒ΄c, βCatalan matrix and related combinatorial identities,β Applied Mathematics and Computation, vol. 215, no. 2, pp. 796β805, 2009.
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