Here and elsewhere it is assumed that the do not take any of the values . The infinite series converges for all when , and for when .
For the function on the right-hand side see §16.2(i).
This notation is from Gasper and Rahman (2004). It is slightly at variance with the notation in Bailey (1935) and Slater (1966). In these references the factor is not included in the sum. In practice this discrepancy does not usually cause serious problems because the case most often considered is .
Here and elsewhere the must not take any of the values , and the must not take any of the values . The infinite series converge when provided that and also, in the case , .
For the function see §16.4(v).
The following definitions apply when and :
The series (17.4.1) is said to be balanced or Saalschützian when it terminates, , , and
The series (17.4.1) is said to be k-balanced when and
The series (17.4.1) is said to be well-poised when and
The series (17.4.1) is said to be nearly-poised when and