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## -pseudo involutions

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**-pseudo involutions**Gi-Sang Cheon, Sung-Tae Jin and Hana Kim Sungkyunkwan University 2009.08.21**Contents**• Riordan group - An involution and pseudo involution - The centralizer of • -pseudo involutions - Classification of -pseudo involutions - Characterization of -pseudo involutions • Application to Toeplitz systems**Riordan group**Riordan group (L. Shapiro, 1991) • with • A Riordan matrix is an infinite lower triangular matrix whose th column has the GF for .**= the set of all Riordan matrices**• forms a group under the Riordan multiplication defined by • is called the Riordan group.**An involution & pseudo involution**• A matrix is called an involution if i.e., • An involution in the Riordan group is called a Riordan involution.**If a Riordan matrix satisfies**i.e., then is called a pseudo involution.**The centralizer of**• The centralizer of in the Riordan group is which is the checkerboard subgroup.**For let**where is a root of i.e.,**Theorem 1.**The centralizer of in the Riordan group is • Note : • is a subgroup of**- pseudo involutions**• For each we say that a Riordan matrix is a - pseudo involution if where if (mod ) otherwise.**Theorem 2.**If is a - pseudo involution then . • Corollary 3. If is a - pseudo involution then**Corollary 4.**If is a - pseudo involution then • Corollary 5. If is a - pseudo involution then the order of is in the Riordan group.**Classification of - pseudo involutions**• = the set of all - pseudo involutions for each • = the collection of ’s • Define a relation on by for iff such that (mod ).**Theorem 6.**The relation is an equivalence relation on • For each it is sufficient to consider - pseudo involutions in the Riordan group.**-sequence of**-sequence of Characterization of - pseudo involutions • Theorem (Rogers ‘78, Sprugnoli ‘94) An i.l.t.m. is a Riordan matrix iff two sequences and with such that**Theorem 7.**is a - pseudo involution with the -seq. GF iff has the -seq. GF where is a root of**-seq. GF =**• A Riordan matrix has a -sequence if**Then is a - pseudo involution**• Theorem 8. Let be a Riordan matrix satisfying where is a root of iff there exists a -sequence GF such that**Example (4-pseudo involution)**Consider the - sequence GF (the GF for twice Fibonacci numbers) Let where satisfies**Theorem 9.**If is a - pseudo involution then is also a - pseudo involution for • Theorem 10. If is a - pseudo involution then is a - pseudo involution for any diagonal matrix**Application to Toeplitz systems**• We define a - pseudo involution of the general linear group by such that for where is the principal submatrix of**Let us consider the problem where**is a Toeplitz matrix.**When is symmetric and positive definite**Toeplitz matrix, there are three algorithms to solve the system : • Durbin’s algorithm • Levinson’s algorithm • Trench’s algorithm**The commutator of**plays an important role to get - pseudo involutions. • Theorem 11. Let Then is a pseudo involution.**Example**Let**The centralizer of in :**• Theorem 12. Let Then if and only if is a - pseudo involution.**Theorem 13.**Let If is a - pseudo involution of Toeplitz type then (mod ).**Theorem 14.**For and let Then is a pseudo involution of Toeplitz type.**Theorem 15.**Let be a Toeplitz matrix. Then is a pseudo involution if and only if the Kronecker product is a - pseudo involution for**Example**Let us consider**Then**is a pseudo involution.