# PDE coupled with the pronic numbers (related to triangular numbers)

I am studying the linear PDE:

$$t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\partial s} \Psi_n(t,s)\tag{1}$$

where the $$b(n)$$ coefficients are $$b(n)=\big\lbrace 2,12,24,40,60,84, \cdot\cdot\cdot\big\rbrace.$$ Here $$b(n)$$ is related to the oblong numbers $$a(n),$$ because $$b(n)=2a(n)$$ for $$n>2.$$

Consider another linear PDE:

$$t\frac{\partial^2}{\partial t^2} \sum_{n=1}^\infty \Phi_n(x,t)=-x\frac{\partial}{\partial x}\sum_{n=1}^\infty \Phi_n(x,t)+\sum_{n=2}^{\infty}a(n)\Phi_n(x,t) \tag{2}$$

So far I have a kernel solution to $$(2)$$ and a kernel solution to $$(1).$$ The solution to $$(1)$$ is:

$$\Psi_n(t,s)=2 \sqrt{\frac{tn}{s}}K_1(2\sqrt{t ns})$$

where $$K_1$$ is a modified Bessel function.

The solution to $$(2)$$ is:

$$\Phi_n(x,t)=e^{\frac{nt}{\log x}}.$$

The PDE's $$(1)$$ and $$(2)$$ are related by the Mellin transform. I took the solution to $$(2)$$ and found it's Mellin transform and then used that to re-construct $$(1)$$.

Are the solutions I found unique? Do PDE's of this type have a name? (i.e. coupled with infinite natural number sequence).

• You are a bit confused because the oblong numbers A002378 $b(n)=n(n+1)=2a(n)$ are twice triangular numbers A000217 $a(n)=n(n+1)/2$. Commented Mar 11 at 1:21
• @Somos No, I am taking the oblong numbers, a(n) and twice the oblong numbers b(n) Commented Mar 16 at 15:24