The Bessel-Maitland function and the generalised Bessel-Maitland function have recently attracted the attention of numerous authors. Integral transformations and fractional calculus results involving these functions have been demonstrated by a number of researchers. The main key of the paper, we establish some finite and infinite integral formulas containing the product of multi-index Bessel Maitland function, incomplete ℵ-function(IAF) and Jacobi polynomial which we have defined in this article. Further, we establish many interesting corollaries by specializing the parameters of IAF in the form of simpler special functions like incomplete H-function, incomplete Wright hypergeometric function, incomplete Fox-Wright function, Fox-Wright function, generalized Mittag-Leffler function etc. with the help of main results.