On Jacobi polynomials and fractional spectral functions on compact symmetric spaces

The Jacobi coefficients cjℓ(α,β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{j}^{\ell }(\alpha ,\beta )$$\end{document} (1≤j≤ℓ;α,β>-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le j\le \ell ; \alpha ,\beta >-1$$\end{document}) associated with the normalised Jacobi polynomials Pk(α,β)(η):=Pk(α,β)(η)/Pk(α,β)(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {P}}_k^{(\alpha , \beta )}(\eta ):=P_{k}^{(\alpha ,\beta )}(\eta )/P_{k}^{(\alpha ,\beta )}(1)$$\end{document} (k≥0;α,β>-1,-1≤η≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 0; \alpha ,\beta >-1, -1\le \eta \le 1$$\end{document}) describe the Maclaurin heat coefficients appearing in the classical Maclaurin expansion of the heat kernel on any N-dimensional compact rank one symmetric space. These coefficients are computed by transforming the even (2ℓ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2\ell )$$\end{document}th (ℓ≥1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell \ge 1)$$\end{document} derivatives of the Jacobi polynomials Pk(α,β)(η)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {P}}_{k}^{(\alpha ,\beta )}(\eta )$$\end{document} into a spectral sum involving the Jacobi operator. In this paper, we generalise this idea by constructing the fractional Taylor heat coefficients (i.e., the coefficients appearing in the fractional Taylor series expansion of the heat kernel) on any rank one symmetric space of compact type. The Riemann–Liouville fractional derivative of normalised Jacobi polynomials Pk(α,β)(η)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {P}}_k^{(\alpha , \beta )}(\eta )$$\end{document} is considered and an interesting spectral identity explicitly described by the fractional Jacobi coefficients is established. The analytical and spectral implications of these fractional coefficients are in turn underlined. The first fractional coefficients are explicitly computed. By extension, fractional Jacobi coefficients play a crucial role in the explicit descriptions of constants appearing in the fractional power series expansion of eigenfunctions involving Jacobi polynomials. We also introduce and construct new zeta functions Zm,ℓμ(α,β)=Zm,ℓμ(α,β)(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {Z}}^{(\alpha ,\beta )}_{m,\ell \mu }={\mathcal {Z}}^{(\alpha ,\beta )}_{m,\ell \mu }(s)$$\end{document}(1≤ℓ≤m,0<μ≤1,s∈C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1\le \ell \le m, 0<\mu \le 1, s\in {\mathbb {C}})$$\end{document} associated with these fractional Taylor heat coefficients. It is interesting to see that this new zeta function can be explicitly described by the newly introduced fractional Minakshisundaram-Pleijel zeta function.