The function transforms input string characters based on their position (odd/even) and case (lowercase/uppercase). If the input contains a digit or a special character, the output is "vank".

[CODE]:
```python
def approx_f(x):
    if any(c.isdigit() or not c.isalnum() for c in x):
        return "vank"
    
    result = []
    for i, c in enumerate(x, start=1):
        if i % 2 == 1:
            if c.islower():
                result.append(chr(((ord(c) - ord('a') + 1) % 26) + ord('a')))
            else:
                result.append(chr(((ord(c) - ord('A') - 1) % 26) + ord('A')))
        else:
            if c.islower():
                result.append(chr(((ord(c) - ord('a') - 1) % 26) + ord('a')))
            else:
                result.append(chr(((ord(c) - ord('A') + 1) % 26) + ord('A')))
    
    return "".join(result)
```
Please note that this code is an approximation of the function behavior, and it may not cover all possible cases.
