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In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek–Jacobi type weightw   (  x  ,  t  )   :  =    (  1  −   x  2   )   α    e   −   t   1  −   x  2,    x  ∈   [  −  1  ,  1  ]   ,    α  >  0  ,    t  >  0  .By using the ladder operator approach, we establish the second‐order difference equations satisfied by the recurrence coefficientβ  n    (  t  )and the sub‐leading coefficient    p  (  n  ,  t  )    of the monic orthogonal polynomials, respectively. We show that the logarithmic derivative ofβ  n    (  t  )can be expressed in terms of a particular Painlevé V transcendent. The large   n   asymptotic expansions ofβ  n    (  t  )and    p  (  n  ,  t  )    are obtained by using Dyson's Coulomb fluid method together with the related difference equations. Furthermore, we study the associated Hankel determinantD  n    (  t  )and show that a quantityσ  n    (  t  )    , allied to the logarithmic derivative ofD  n    (  t  )    , can be expressed in terms of the   σ  ‐function of a particular Painlevé V. The second‐order differential and difference equations forσ  n    (  t  )are also obtained. In the end, we derive the large   n   asymptotics ofσ  n    (  t  )andD  n    (  t  )from their relations withβ  n    (  t  )and    p  (  n  ,  t  )   .

Differential, difference, and asymptotic relations for Pollaczek–Jacobi type orthogonal polynomials and their Hankel determinants