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Abstract
In this paper, we investigate degenerate versions of the generalized pth order Franel numbers which are certain finite sums involving powers of binomial coefficients. In more detail, we introduce degenerate generalized hypergeometric functions and study degenerate hypergeometric numbers of order p. These numbers involve powers of λ-binomial coefficients and λ-falling sequence, and can be represented by means of the degenerate generalized hypergeometric functions. We derive some explicit expressions and combinatorial identities for those numbers. We also consider several related special numbers like λ-hypergeometric numbers of order p and Apostol type λ-hypergeometric numbers of order p, of which the latter reduce in a limiting case to the generalized pth order Franel numbers.
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Details
1 Xian Technological University, School of Science, Xian, China (GRID:grid.460183.8) (ISNI:0000 0001 0204 7871); Kwangwoon University, Department of Mathematics, Seoul, Republic of Korea (GRID:grid.411202.4) (ISNI:0000 0004 0533 0009)
2 Sogang University, Department of Mathematics, Seoul, Republic of Korea (GRID:grid.263736.5) (ISNI:0000 0001 0286 5954)
3 Kwangwoon University, Department of Mathematics, Seoul, Republic of Korea (GRID:grid.411202.4) (ISNI:0000 0004 0533 0009)
4 Gyeongsang National University, Department of Mathematics Education and ERI, Jinju, Republic of Korea (GRID:grid.256681.e) (ISNI:0000 0001 0661 1492)