The generalized hypergeometric function \(_{p}F_{q}\) is defined for complex parameters and argument by the series \[ _{p}F_{q} ( (a_{p}); (b_{q}) |x)= \sum_ {k=0}^{\infty} \frac {((a_{p}))_{k}}{((b_{q}))_{k}} \frac{x^{k}}{k!}, \]
where \((a_{p})= (a_{1}, \dotsc , a_{p})\) as well as \( ((a_{p}))_{k}= (a_{1})_{k}(a_{2})_{k} \dotsm (a_{p})_{k}\) denotes the Pochhammer symbol for the vector \((a_{p})\). Notice that this series converges for \(|x|< \infty\) when \(q\geq p\), while it converges in \(|x|<1\) when \(q=p-1\). Notice that when one of the parameters \(a_{j}\) in the numerator is a negative integer or zero, the above series become a polynomial of degree \(-a_{j}\) in the variable \(x\).
In this contribution, the authors deal with transformation formulas for the hypergeometric functions \(_{r+2}F_{r+1}\) and \(_{r+1 }F_{r+1},\) respectively, whose numerator and denominator parameters differ by positive integers \((m_{r})\). Indeed, they prove that
\[ _{r+2}F_{r+1} (a, b, (f_{r}+ m_{r}); c, (f_{r})| x)= (1-x)^{-a} \,\;_{m+2} F_{m+1} (a, \lambda, (\xi_{m}+1); c, (\xi_{m}) |\frac{x}{x-1}),\tag{i} \] for \(|x|<1,\;\operatorname{Re}\;x <1/2\). \[ _{r+2}F_{r+1} (a, b, (f_{r}+ m_{r}); c, (f_{r})| x)= (1-x)^{c-a-b-m} \,\;_{m+2} F_{m+1} (\lambda, \lambda ', (\eta_{m}+1); c, (\eta_{m})| x ),\tag{ii} \] for \(|x|<1\). \[ _{r+1}F_{r+1} (b, (f_{r}+ m_{r}); c, (f_{r})| x)= e^{x} \,\;_{m+1} F_{m+1} (\lambda, (\xi_{m}+1); c, (\xi_{m}) | -x),\tag{iii} \] for \(|x|<\infty\).
In these transformation formulas, \(m= \sum_{k=1}^{r} m_{k}\), \(\lambda= c-b- m \), \(\lambda ' = c- a- m\). The entries of the vectors \((\xi_{m})\) and \((\eta_{m})\) are non vanishing zeros of certain associated parametric polynomials of degree \(m\), denoted by \(Q_{m}(x)\), assuming certain restrictions on some of the parameters of the generalized hypergeometric functions on both sides of (i)–(iii) (see A. R. Miller [J. Comput. Appl. Math. 231, No. 2, 964–972 (2009; Zbl 1221.33011)]).
When \((m_{r})\) is empty, i.e., \(m=0\), (i)–(ii) become Euler’s classical first and second transformations for the Gauss hypergeometric function while (iii) yields Kummer’s transformation formula for the confluent hypergeometric function. If \(m_{1}= \dotsb = m_{r}=1\), transformation formulas (i)–(ii) have been analyzed in [A. R. Miller and R. B. Paris, Z. Angew. Math. Phys. 62, No. 1, 31–45 (2011; Zbl 1225.33005)], while (iii) was obtained in [A. R. Miller, loc. cit.].
The explicit expression of the polynomial \(Q_{m}(x)\) for the above transformations is deduced. On the other hand, for \(_{r+2}F_{r+1}\), two generalizations of the well-known quadratic transformation formulas for the Gauss hypergeometric functions are given. Here parameter sequences in numerator and denominator are, again, zeros of a polynomial \(Q_{m}(x)\) whose explicit expression is done.
Some examples of the above results are shown when \(r=2\) for particular choices of \((m_{1},m_{2})\).