  A novel family of $-1$ orthogonal polynomials called the Chihara polynomials
is characterized. The polynomials are obtained from a "continuous" limit of the
complementary Bannai-Ito polynomials, which are the kernel partners of the
Bannai-Ito polynomials. The three-term recurrence relation and the explicit
expression in terms of Gauss hypergeometric functions are obtained through a
limit process. A one-parameter family of second-order differential Dunkl
operators having these polynomials as eigenfunctions is also exhibited. The
quadratic algebra with involution encoding this bispectrality is obtained. The
orthogonality measure is derived in two different ways: by using Chihara's
method for kernel polynomials and, by obtaining the symmetry factor for the
one-parameter family of Dunkl operators. It is shown that the polynomials are
related to the big $-1$ Jacobi polynomials by a Christoffel transformation and
that they can be obtained from the big $q$-Jacobi by a $q\rightarrow -1$ limit.
The generalized Gegenbauer/Hermite polynomials are respectively seen to be
special/limiting cases of the Chihara polynomials. A one-parameter extension of
the generalized Hermite polynomials is proposed.
