A new recurrence relation for the truncated very-well-poised ${}_6{\psi }_6$ series and Bailey's summation formula

In this paper we introduce the so-called truncated very-well-poised ${}_6{\psi }_6$ series and set up an explicit recurrence relation for it by means of the classical Abel lemma on summation by parts. This new recurrence relation implies an elementary proof of Bailey’s well-known ${}_6{\psi }_6$ summation formula.