Looking for more properties & references for what I call backward harmonic functions

Question

While analyzing a discrete version if Ito's Lemma, I became interested in a family of functions defined by the following:

$$ f(n, k) = \frac{f(n+1, k+1) + f(n+1, k-1)}{2} \tag{$1$} $$

If $f$ satisfies (1), then we call $f$ a backwards harmonic function. For example, $f(k, n) = k^2 - n$ satisfies the above, as does $f(n, k) = k$ and $f(n, k) = \text{const}$.

I've been able to prove the following about backward harmonic functions:

1. Extreme values: If $f$ satisfies equation (1), then $f$ has its maximum value and minimum value on the boundary ${(n, ·)}$.

2. Vector space: If $g$ and $h$ are backwards harmonic functions and $a, b \in R$, then $a · g + b · h$ is also a backward harmonic function.

3. Uniqueness: If $f$ and $g$ are backwards harmonic and $f(N, ·) = g(N, ·)$, then $f(m, k) = g(m, k), 0 \le m \le N, \forall k$

4. Suppose $f(n, k)$ is backwards harmonic and satisfies $f(n + 1, k) = f(n, k)$ for all $n, k$, then $f(n, k) = (f_0 − f(n, −1))\cdot k + f_0$ for any $n$

What more can we learn about these functions? Are they already studied under a different name?

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The reason I am interested in these functions is related to another theorem I have proved. If $B_n$ is a random walk defined by $B_0=0$ and $P(\Delta B_n = 1) = P(\Delta B_n = -1) = 0.5$, then

5. $f$ is a backwards harmonic function iff $f(n, B_n)$ is a martingale with respect to $B_n$.

6. If $f$ is a backwards harmonic function, then $E[|f(n + 1, B_{n+1})|] \ge E[|f(n, B_n)|]$

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Edit 1.

The question When is a polynomial of an unbiased random walk a martingale? gives other polynomials, which generally seem to be related to Hermite polynomials. (Correction: see Edit 2.)

There are other, non-polynomials that satisfy this that I've found:

$$ f(n,k) = \begin{cases} 1,& \text{if } n = 0\\ 2^{n-1}, & n > 0 \;\text{and}\; (k = n \;\text{or}\; k = -n) \\ 0, & \text{otherwise} \end{cases} $$

00                            [1]                             sum=1
01                           [1, 1]                           sum=2
02                         [2, 0, 2]                          sum=4
03                        [4, 0, 0, 4]                        sum=8
04                      [8, 0, 0, 0, 8]                       sum=16
05                    [16, 0, 0, 0, 0, 16]                    sum=32
06                  [32, 0, 0, 0, 0, 0, 32]                   sum=64
07                 [64, 0, 0, 0, 0, 0, 0, 64]                 sum=128
08              [128, 0, 0, 0, 0, 0, 0, 0, 128]               sum=256
09             [256, 0, 0, 0, 0, 0, 0, 0, 0, 256]             sum=512
10           [512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 512]            sum=1024

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Edit 2.

There is a family of polynomial solutions that look like Hermite polynomials, but are different (perhaps a discrete form of them?). Define $$ f_0(n,k) = 1, \\f_1(n, k) = k $$

Define

$$ f_m(n, k) = kf_{m-1}(n, k) + n\sum_{j=1}^{\frac{m}{2}} (-1)^j \cdot \text{Tan}(j)\cdot{m-1\choose 2j-1}\cdot f_{m - 2j}(n,k) $$

where $\text{Tan}(n)$ is a tangent number. I claim that $f_m(n,k)$ is also backwards harmonic. This generates the sequence of polynomials as in When is a polynomial of an unbiased random walk a martingale?

$$ \begin{align} f_2(n, k) &= k^{2} - n\\ f_3(n, k) &= k^{3} - 3 k n\\ f_4(n, k) &= k^{4} - 6 k^{2} n + 3 n^{2} + 2 n\\ f_5(n, k) &= k^{5} - 10 k^{3} n + 15 k n^{2} + 10 k n\\ f_6(n, k) &= k^{6} - 15 k^{4} n + 45 k^{2} n^{2} + 30 k^{2} n - 15 n^{3} - 30 n^{2} - 16 n\\ f_7(n, k) &= k^{7} - 21 k^{5} n + 105 k^{3} n^{2} + 70 k^{3} n - 105 k n^{3} - 210 k n^{2} - 112 k n \end{align} $$

Some properties:

2. $f_m(n, x + y) = \sum_{i=0}^m {m\choose i}f_{m-i}(n, x) y^{i}$
3. Equivalent to above: $\frac{d}{dx}f_m(n, x) = m f_{m-1}(n, x)$
4. $f_m(n+1, k)$ looks to be related to the Swiss-Knife polynomials