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### Recurrences for Callan's Generalization of Narayana Polynomials

CHEN Xi1, YANG Arthur Li Bo2, ZHAO James Jing Yu3

1. 1. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China;
2. Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, China;
3. School of Mathematics, Tianjin University, Tianjin 300350, China
• 收稿日期:2020-09-10 修回日期:2020-12-04 出版日期:2022-08-25 发布日期:2022-08-02
• 通讯作者: ZHAO James Jing Yu,Email:jjyzhao@tju.edu.cn
• 作者简介:CHEN Xi,Email:chenxi@dlut.edu.cn;YANG Arthur Li Bo,Email:yang@nankai.edu.cn
• 基金资助:
CHEN was supported by the National Natural Science Foundation of China under Grant No. 11601062. YANG was supported in part by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China under Grant Nos. 11522110 and 11971249, respectively. ZHAO was partially supported by the National Natural Science Foundation of China under Grant Nos. 11771330 and 11971203.

CHEN Xi, YANG Arthur Li Bo, ZHAO James Jing Yu. Recurrences for Callan's Generalization of Narayana Polynomials[J]. 系统科学与复杂性, 2022, 35(4): 1573-1585.

CHEN Xi, YANG Arthur Li Bo, ZHAO James Jing Yu. Recurrences for Callan's Generalization of Narayana Polynomials[J]. Journal of Systems Science and Complexity, 2022, 35(4): 1573-1585.

### Recurrences for Callan's Generalization of Narayana Polynomials

CHEN Xi1, YANG Arthur Li Bo2, ZHAO James Jing Yu3

1. 1. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China;
2. Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, China;
3. School of Mathematics, Tianjin University, Tianjin 300350, China
• Received:2020-09-10 Revised:2020-12-04 Online:2022-08-25 Published:2022-08-02
• Contact: ZHAO James Jing Yu,Email:jjyzhao@tju.edu.cn
• Supported by:
CHEN was supported by the National Natural Science Foundation of China under Grant No. 11601062. YANG was supported in part by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China under Grant Nos. 11522110 and 11971249, respectively. ZHAO was partially supported by the National Natural Science Foundation of China under Grant Nos. 11771330 and 11971203.
By using Chen, Hou and Mu's extended Zeilberger algorithm, the authors obtain two recurrence relations for Callan's generalization of Narayana polynomials. Based on these recurrence relations, the authors further prove the real-rootedness and asymptotic normality of Callan's Narayana polynomials.
By using Chen, Hou and Mu's extended Zeilberger algorithm, the authors obtain two recurrence relations for Callan's generalization of Narayana polynomials. Based on these recurrence relations, the authors further prove the real-rootedness and asymptotic normality of Callan's Narayana polynomials.
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