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Our aim is to investigate through a special family of Laguerre–Hahn forms of class one verifying the three-term recurrence relation withβ  n   =(   -  1   )   n   ⁢   β  0${\beta_{n}=(-1)^{n}\beta_{0}}$   ,n  ≥  0${n\geq 0}$   , by using quadratic decomposition. The corresponding characteristic elements of the three-term recurrence relation are given explicitly.

Characterizations of a special family of Laguerre–Hahn forms of class one