How can I translate a MathWorld statement into Mathematica syntax?

Waiting for a more elegant solution, you can try this:

Creates a differential operator from a polynomial in x:

DiffOp[poly_] /; PolynomialQ[poly, x] := Block[{Dop},
  Dop[c_, {ord_}] := c*(D[#, {x, ord - 1}] &);
  Return[Total[MapIndexed[Dop, CoefficientList[poly, x]]]];
  ];

For instance:

op = DiffOp[ChebyshevT[4, x]]

FullSimplify[op[BesselI[0, x]]]

BesselI[4, x]

---

Update: maybe slightly better (and without Return[], see Carl Woll remark):

DiffOp[poly_, x_, f_] /; PolynomialQ[poly, x] := 
 CoefficientList[poly, x].Table[D[f, {x, i}], {i, 0, Exponent[poly, x] }]

For instance:

DiffOp[ChebyshevT[4, x], x, f[x]]

prints: $$ 8 f^{(4)}(x)-8 f''(x)+f(x) $$

and

FullSimplify[DiffOp[ChebyshevT[4, x], x, BesselI[0,x]]]

returns

Bessel[4,x]

Assuming the idea is to translate the derivative operator (which must make this Q&A a duplicate, no?), this was what I came up with...

ClearAll[CircleDot];
(* assumes op is a polynomial with constant coefficients and the variable in f is x *)
CircleDot[op_, f_] /; PolynomialQ[op, Derivative[1]] && FreeQ[op, x] :=
   With[{c = CoefficientList[op, Derivative[1]]},
    c.NestList[D[#, x] &, f, Length@c - 1]
   ];

ChebyshevT[3, Derivative[1]] \[CircleDot] BesselI[0, x] // Expand
(*  BesselI[3, x]  *)

You can make use of my DifferentialOperator paclet to do this. Install with:

PacletInstall["https://github.com/carlwoll/DifferentialOperator/releases/download/0.1/DifferentialOperator-0.0.1.paclet"]

and load with:

<<DifferentialOperator`

Then (I had to use an image since the partial derivative in the input doesn't translate well to MSE):