It is well known that there is no general formula for the roots of a polynomial of degree $\geq$ 5 solely in terms of arithmetical operations and radicals.
Are there formulas using other kinds of operations or transcendental functions of coefficients? Even just for quintics?
For quintics the question can also be asked as: is there a formula for the roots of $z^5 + a z + 1$ in terms of $a$?
There are plenty of such formulas. For example, for the quintic $z^5-z+a=0$, the Burmann-Lagrange formula gives $$z=-\sum_{k=0}^\infty{5k\choose k}\frac{a^{4k+1}}{4k+1}.$$ There are also formulas based on theta-functions; see, for example H. Umemura, Solution of algebraic equations using theta-constants, Appendix to the book D. Mumford, Tata Lectures on Theta, vol. II. They solve algebraic equations of any degree.
Remark. When we are talking of formulas involving infinite processes, like series or integrals, he difference between a "formula" and an "algorithm" essentially disappears. There is a famous algorithm which solves polynomial equations (and many other equations) which is called Newton's method. It gives you faster convergence than most formulas.
A trinomial root, including non-integer powers, can be found by means of the confluent Fox-Wright $_{1}\Psi_{1}$ function, a generalization of the confluent hypergeometric function $_{1}F_{1}$, defined as $$_{1}\Psi_{1}([a,\alpha];[b,\beta];\zeta) = \sum_{n=0}^{\infty}\frac{\Gamma(a+n\alpha)}{\Gamma(b+n\beta)}\frac{\zeta^n}{n!}$$ where $-a-n\alpha\notin\mathbb{N}_0$ for $n\in\mathbb{N}_0$ and $\zeta\in\mathbb{C}$. It is convergent for $|\zeta|<|\alpha^{-\alpha}\beta^\beta|$.
For $\gamma>1$ the trinomial equation (some algebra steps can set a general trinomial equation in this form) $$z-\omega z^\gamma-1=0$$ has this closed form root $$z =\,_{1}\Psi_{1}([1,\gamma];[2,\gamma-1];\omega)$$ which can be set as $$z=1+\sum_{n=1}^\infty\binom{n\gamma}{n-1}\frac{\omega^n}{n}$$ whose convergence region is $|\omega|<|(\gamma-1)^{\gamma-1}\gamma^{-\gamma}|$. For non integer $\gamma>1$ binomials must be interpreted in terms of $\Gamma$ function.
References
Miller A.R., Moskowitz I.S. Reduction of a Class of Fox-Wright Psi Functions for Certain Rational Parameters. Computers Math. Applic. Vol. 30, No. 11, pp. 73-82, (1995). Pergamon
Dzevad Belkic. All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox-Wright function: illustration for genome multiplicity in survival of irradiated cells. Journal of Mathematical Chemistry (2019) 57:59-106
Note
Wolfram Mathematica 12.3 has incorporated recently Fox-H Function (as an experimental feature). Generalized Fox-Wright Function is a Fox-H's special case.
