Heat kernel expansion for higher order minimal and nonminimal operators

We build a systematic calculational method for the covariant expansion of two-point heat kernel $\stackrel{^}{K}(\ensuremath{\tau}|x,{x}^{\ensuremath{'}})$ generic minimal and nonminimal differential operators any order. This is in powers dimensional background field objects---the coefficients operator corresponding spacetime vector bundle curvatures, suitable renormalization effective theory applications. For whose principal symbol given by an arbitrary power Laplacian $(\ensuremath{-}\ensuremath{\square}{)}^{M}$, $M>1$, this result generalizes well-known Schwinger--DeWitt (or Seeley--Gilkey) to infinite series positive negative fractional proper time ${\ensuremath{\tau}}^{1/M}$, weighted generalized exponential functions dimensionless argument $\ensuremath{-}\ensuremath{\sigma}(x,{x}^{\ensuremath{'}})/2{\ensuremath{\tau}}^{1/M}$ depending on Synge world function $\ensuremath{\sigma}(x,{x}^{\ensuremath{'}})$. The are determined chain auxiliary acting parallel transport tensor, which their turn follow from solution special recursive equations. derivation these equations based Fourier transform curved spacetime. $\ensuremath{\tau}$ vanishes coincidence limit ${x}^{\ensuremath{'}}=x$, makes proposed consistent with Seeley--Gilkey it beyond diagonal form asymptotic domain $\ensuremath{\sigma}(x,{x}^{\ensuremath{'}})\ensuremath{\ll}{\ensuremath{\tau}}^{1/M}$, $\ensuremath{\tau}\ensuremath{\rightarrow}0$. Consistency also checked verification known results second-order extension fourth-order operator. Possible improvement suggested approach noncommutative algebra $\ensuremath{\square}$ universal functional traces briefly discussed.