Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $[T(n,k)]_{n,k\geqslant0}$ be a triangle of positive numbers satisfying the three-term recurrence relation \[ T(n,k)=(a_1n+a_2k+a_3)T(n-1,k)+(b_1n+b_2k+b_3)T(n-1,k-1). \] ,  give a new sufficient condition for linear transformations \[ Z_n(q)=\sum_{k=0}^{n}T(n,k)X_k(q) \] that preserves the strong $q$-log-convexity of polynomials sequences. As applications,  show linear transformations, given by matrices of the binomial coefficients, the Stirling numbers of the first kind and second kind, the Whitney numbers of the first kind and second kind, preserving the strong $q$-log-convexity in a unified manner.
['Lily Li Liu', 'Ya-Nan Li']
I am writing a research paper on a idea Recurrence Relations for the Linear Transformation Preserving the Strong $q$-Log-Convexity
[{"from": "human", "value": "I am writing a research paper on a idea Recurrence Relations for the Linear Transformation Preserving the Strong $q$-Log-Convexity"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $[T(n,k)]_{n,k\\geqslant0}$ be a triangle of positive numbers\u00a0satisfying the three-term recurrence relation \\[ T(n,k)=(a_1n+a_2k+a_3)T(n-1,k)+(b_1n+b_2k+b_3)T(n-1,k-1). \\] ,  give a new sufficient condition for linear transformations \\[ Z_n(q)=\\sum_{k=0}^{n}T(n,k)X_k(q) \\] that preserves the strong $q$-log-convexity of polynomials sequences.\u00a0As applications,  show linear transformations, given by matrices of the binomial coefficients,\u00a0the Stirling numbers of the first kind and second kind, the Whitney numbers of the first kind and second kind,\u00a0preserving the strong $q$-log-convexity in a unified manner."}]