Hankel Determinants Of Q-analogue Of Several Kinds Of Combinatorial Numbers

Hankel matrix is a very important research object in the theory of nonnegative matrices.The calculation of Hankel determinants is also an significant topic in combinatorics,which often arises not only in combinatorics,but also in other mathematical fields such as signal digital processing,mechanics,numerical computation and system control.Although many achievements have been made in the calculation of Hankel determinants of combinatorial numbers,their calculation is often difficult.In this dissertation,we mainly evaluate some Hankel determinants of q-analogue of Catalan numbers,Motzkin numbers and Schr(?)der numbers.The details of this dissertation are as follow:The first part mainly discusses the problems associated with the combinatorial number c(n,m,a,b)as defined by Cigler.Firstly,its q-analogue c(n,m,a,b,q)is defined,and the combinatorial interpretation of c(n,m,a,b,q)is explained by using lattice paths.Secondly,the values of Hankel determinants of c(n,m,a,b,q)are given in some specific cases by using Lindstrom-Gessel-Viennot lemma and the relations between the shifted Hankel determinants are given.Finally,we also use algebraic methods to give another proof of the above relations and to prove the relations between the other shifted Hankel determinants of c(n,m,a,b,q).In the second part,we mainly use the lattice paths method to calculate the Hankel determinants of q-Catalan numbers and q-Motzkin numbers.The third part mainly uses the lattice paths method to calculate the q-analogue of Hankel determinants of Catalan numbers and Motzkin numbers,which also confirms two conjectures proposed by Masao Ishikawa.Similarly,we also define a class of q-analogue of Schr(?)der numbers and calculate the q-analogue of Hankel determinants of Schr(?)der numbers.