Construction and calculation of reproducing kernel determined by various linear differential operators

We presented a method to construct and calculate the reproducing kernel for the linear differential operator with constant coefficients and a single latent root; further, we gave the formula for calculation. Additionally, by studying the recurrence relation of the reproducing kernel with arithmetic latent roots, we found that the reproducing kernel with a multi-knots interpolation constraint can be concisely represented by one with an initial-value constraint.